Number Theoretic Considerations and Advanced Analysis of
Autoregressive Models
Proof for Theorem 11.3.2
Analysis of 𝑔𝑘(𝑠) and ℎ(𝜃)
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
• Goal: Determine when 𝑔𝑘(𝑠) is positive for infinitely many 𝑘’s
• Define function ℎ(𝜃) by substituting 𝑘𝜙 for 𝜃 in 𝑔𝑘
• Introduce constants: 𝐴 = (𝑠 −1)2cos𝜙, 𝐵 = (𝑠2−1)sin𝜙, 𝐶 = 2𝑠sin2𝜙cos𝜙
Lemma 11.3.6: Properties of ℎ(𝜃)
Key results:
• ℎ(𝜃) is maximal at 𝜃∗, unique modulo 𝜋
• cos(2𝜃∗)= 𝐴/√𝐴2+𝐵2, sin(2𝜃∗)= 𝐵/√𝐴2+ 𝐵2
• Maximum value: ℎ(𝜃∗)= −𝐴
2+√𝐴2+𝐵2
2−𝐶
Lemma 11.3.7: Positivity of ℎ(𝜃)
In the domain 𝑎 < 0, 𝑏 < −1, and 𝑏 < −𝑎2/4:
• ℎ(𝜃) has a positive supremum if and only if 𝑏 < 𝑎 − 1
Lemma 11.3.8: Behavior of 𝑔𝑘(𝑟2)
Assume 𝑎 < 0 and 𝑏 < min(−𝑎2/4,−1):
• If 𝑏 < 𝑎 − 1 and 𝜙 is an irrational multiple of 2𝜋: limsup𝑘→∞𝑔𝑘(𝑟2)> 0
• If 𝑏 > 𝑎 − 1: ∃𝜖 > 0 such that 𝑔𝑘(𝑟2)≤ −𝜖 for all 𝑘 ≥ 1
Conclusion of Theorem 11.3.2 Proof
• For 𝑏 < 𝑎 − 1: (𝑎,𝑏) is covered by infinitely many regions 𝑅𝑘
• For 𝑏 > 𝑎 − 1: (𝑎,𝑏) is covered by at most a finite number of regions 𝑅𝑘
• The complement of the potentially uncovered set is dense in 𝑏 < 𝑎 − 1
Additional Insights
Behavior when 𝑎 < 0 and 𝑏 > −𝑎2/4
• No region 𝑅𝑘 covers (𝑎,𝑏)
• Sequence 𝑘 ↦ 𝐶𝑛−𝑘,𝑛−𝑘 is decreasing
Expression for 𝑐(𝑎,𝑏)
For 𝑎 < 0 and 𝑏 > −𝑎2/4:
𝑐(𝑎,𝑏)= 𝐾𝑠,𝛼
2𝛼 ∑ |𝐴𝑛−𝑗,1|𝛼
1≤𝑗≤𝑛−1
Behavior for 𝑎 < 0 and 𝑎 −1 < 𝑏 < −𝑎2/4
• At most a finite number of regions 𝑅𝑘 cover (𝑎,𝑏)
• Conjecture: No region 𝑅𝑘 covers such pair (𝑎,𝑏)
Number Theoretic Considerations
• For 𝜙 ∈ (0,𝜋/2): sin𝑘𝜙sin(𝑘 −1)𝜙 contains infinitely many positive values
• When 𝜙 is a rational multiple of 𝜋: (𝑘𝜙)𝑘≥1 is periodic modulo 2𝜋
Conjecture and Final Remarks
• Conjecture: Theorem 11.3.2 is sharp, i.e., the described region coincides with ⋃𝑘≥1
𝑅𝑘
• Complete description of behavior in the stability region is possible using Lemmas
11.3.4 and 11.3.5
•
