r t r s tr st t s
play

r tr str stts - PowerPoint PPT Presentation

Jan 25, 2024 •348 likes •955 views

r tr str stts ttrs t epn Starosta Faculty of Information Technology Czech Technical University in Prague AAMP13 2016, Prague

  1. ◆✉♠❜❡r t❤❡♦r② ❛♥❞ s♣❡❝tr❛❧ ✐♥st❛❜✐❧✐t✐❡s ✐♥ ♠❡t❛♠❛t❡r✐❛❧s Štˇ epán Starosta Faculty of Information Technology Czech Technical University in Prague AAMP13 2016, Prague 1 / 16

  2. Motivation Markov constant Results ❚❤❡ s❡t S ( α ) When analyzing the spectrum of the operator � = ∂ 2 ∂ x 2 − ∂ 2 � f ( x , y ) = λ f ( x , y ) , ∂ y 2 , f | ∂ R = 0 , with R being a rectangle with sides a and b : a R b 2 / 16

  3. Motivation Markov constant Results ❚❤❡ s❡t S ( α ) When analyzing the spectrum of the operator � = ∂ 2 ∂ x 2 − ∂ 2 � f ( x , y ) = λ f ( x , y ) , ∂ y 2 , f | ∂ R = 0 , with R being a rectangle with sides a and b : a R b the following set comes up � � � � k m 2 S ( α ) = set of all accumulation points of m − α : k , m ∈ Z , where α = a b ∈ R . [Talk of D. Krejˇ ciˇ rík] 2 / 16

  4. Motivation Markov constant Results ❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s For α ∈ R , the continued fraction expansion of α is the sequence ( a i ) i = 0 with a i ∈ Z , a j > 0 for j > 0 such that 1 α = a 0 + . 1 a 1 + 1 a 2 + a 3 + · · · Notation: α = [ a 0 , a 1 , a 2 , . . . ] . 3 / 16

  5. Motivation Markov constant Results ❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s For α ∈ R , the continued fraction expansion of α is the sequence ( a i ) i = 0 with a i ∈ Z , a j > 0 for j > 0 such that 1 α = a 0 + . 1 a 1 + 1 a 2 + a 3 + · · · Notation: α = [ a 0 , a 1 , a 2 , . . . ] . a i is called a partial coefficient . 3 / 16

  6. Motivation Markov constant Results ❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s For α ∈ R , the continued fraction expansion of α is the sequence ( a i ) i = 0 with a i ∈ Z , a j > 0 for j > 0 such that 1 α = a 0 + . 1 a 1 + 1 a 2 + a 3 + · · · Notation: α = [ a 0 , a 1 , a 2 , . . . ] . a i is called a partial coefficient . The N -th convergent of α is 1 p N = a 0 + ∈ Q . q N 1 a 1 + 1 a 2 + 1 ... + a N 3 / 16

  7. Motivation Markov constant Results ▼❛r❦♦✈ ❝♦♥st❛♥t m 2 � k � � � S ( α ) = set of all accumulation points of m − α : k , m ∈ Z The Markov constant of α is if α �∈ Q m → + ∞ m � m α � = min |S ( α ) | , µ ( α ) = lim inf where � x � is the distance to the nearest integer � x � = min {| x − n | : n ∈ Z } 4 / 16

  8. Motivation Markov constant Results ▼❛r❦♦✈ ❝♦♥st❛♥t m 2 � k � � � S ( α ) = set of all accumulation points of m − α : k , m ∈ Z The Markov constant of α is if α �∈ Q m → + ∞ m � m α � = min |S ( α ) | , µ ( α ) = lim inf where � x � is the distance to the nearest integer � x � = min {| x − n | : n ∈ Z } L = { µ ( α ): α ∈ R } ( L is sometimes called the Lagrange spectrum) 4 / 16

  9. Motivation Markov constant Results ❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L Hurwitz’s theorem: there exists infinitely many p , q ∈ Z such that � � � α − p 1 � � √ � < � � q 5 q 2 (also follows from an earlier result of Markov) 5 / 16

  10. Motivation Markov constant Results ❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L Hurwitz’s theorem: there exists infinitely many p , q ∈ Z such that � � � α − p 1 � � √ � < � � q 5 q 2 (also follows from an earlier result of Markov) 1 thus, µ ( α ) ≤ √ 5 5 / 16

  11. Motivation Markov constant Results ❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L Hurwitz’s theorem: there exists infinitely many p , q ∈ Z such that � � � α − p 1 � � √ � < � � q 5 q 2 (also follows from an earlier result of Markov) 1 thus, µ ( α ) ≤ √ 5 √ √ if α is not equivalent to 1 + 5 = [ 1 , 1 , 1 , . . . ] , then we may replace 5 √ 2 by 8 5 / 16

  12. Motivation Markov constant Results ❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L Hurwitz’s theorem: there exists infinitely many p , q ∈ Z such that � � � α − p 1 � � √ � < � � q 5 q 2 (also follows from an earlier result of Markov) 1 thus, µ ( α ) ≤ √ 5 √ √ if α is not equivalent to 1 + 5 = [ 1 , 1 , 1 , . . . ] , then we may replace 5 √ 2 by 8 √ if, moreover, α is not equivalent to 1 + 2 = [ 2 , 2 , 2 , . . . ] , then we √ √ 221 may replace 8 by 5 5 / 16

  13. Motivation Markov constant Results ❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ ( α ) Let p N q N be the convergents of α = [ a 0 , a 1 , a 2 , . . . ] , then � � α − p N 1 q 2 = ( − 1 ) N [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] . N q N 6 / 16

  14. Motivation Markov constant Results ❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ ( α ) Let p N q N be the convergents of α = [ a 0 , a 1 , a 2 , . . . ] , then � � α − p N 1 q 2 = ( − 1 ) N [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] . N q N µ ( α ) = lim inf m → + ∞ m � m α � 1 µ ( α ) = lim inf [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] N → + ∞ 6 / 16

  15. Motivation Markov constant Results ❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ ( α ) Let p N q N be the convergents of α = [ a 0 , a 1 , a 2 , . . . ] , then � � α − p N 1 q 2 = ( − 1 ) N [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] . N q N µ ( α ) = lim inf m → + ∞ m � m α � 1 µ ( α ) = lim inf [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] N → + ∞ √ � 1 + � 5 1 1 µ = = √ √ √ 1 + + 1 + 2 5 5 5 − 1 2 2 6 / 16

  16. Motivation Markov constant Results ❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ ( α ) Let p N q N be the convergents of α = [ a 0 , a 1 , a 2 , . . . ] , then � � α − p N 1 q 2 = ( − 1 ) N [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] . N q N µ ( α ) = lim inf m → + ∞ m � m α � 1 µ ( α ) = lim inf [ a N + 1 , a N + 2 , . . . ] + [ 0 , a N , . . . , a 1 ] N → + ∞ √ � 1 + � 5 1 1 µ = = √ √ √ 1 + + 1 + 2 5 5 5 − 1 2 2 √ 1 1 1 � � µ 1 + = √ √ 2 − 2 = √ 2 = √ 2 1 + 2 + 1 + 2 8 6 / 16

  17. Motivation Markov constant Results ❚❤❡ s❡t L 0 7 / 16

  18. Motivation Markov constant Results ❚❤❡ s❡t L 0 1 √ 5 7 / 16

  19. Motivation Markov constant Results ❚❤❡ s❡t L 1 √ 8 0 1 √ 5 1 √ 8 7 / 16

  20. Motivation Markov constant Results ❚❤❡ s❡t L 5 √ 221 1 √ 8 0 1 √ 5 5 √ 221 1 √ 8 7 / 16

  21. Motivation Markov constant Results ❚❤❡ s❡t L 5 √ 221 13 1 √ √ 1571 8 0 1 √ 5 5 √ 221 13 1 √ √ 1571 8 7 / 16

  22. Motivation Markov constant Results ❚❤❡ s❡t L largest accumulation point 5 √ 221 13 1 √ √ 1571 8 0 1 1 √ 3 5 5 √ 221 13 1 √ √ 1571 8 1 7 / 16

  23. Motivation Markov constant Results ❚❤❡ s❡t L largest accumulation point 5 √ 221 13 1 √ √ 1571 8 0 1 1 √ 3 discrete part 5 7 / 16

  24. Motivation Markov constant Results ❚❤❡ s❡t L largest accumulation point 5 √ 221 13 1 √ √ 1 1571 8 0 1 1 √ √ 12 3 discrete part 5 partial coefficients in { 1 , 2 } 7 / 16

  25. Motivation Markov constant Results ❚❤❡ s❡t L largest accumulation point 5 √ 221 13 1 √ √ 1 1 1571 8 0 1 1 √ √ √ 13 12 3 discrete part 5 at least one partial coefficient larger than 2 partial coefficients in { 1 , 2 } 7 / 16

  26. Motivation Markov constant Results ❚❤❡ s❡t L largest accumulation point 5 √ 221 13 1 √ √ 1 1 1571 8 0 F 1 1 √ √ √ 13 12 3 discrete part 5 at least one partial coefficient larger than 2 partial coefficients in { 1 , 2 } √ 555391024 − 70937 462 ≈ 0 . 22 F = 2507812168 (Freiman’s constant) 7 / 16

  27. Motivation Markov constant Results ❚❤❡ s❡t L largest accumulation point 5 √ 221 13 1 √ √ 1 1 1571 8 0 F 1 1 √ √ √ 13 12 3 discrete part 5 continuous part (Hall’s ray) at least one partial coefficient larger than 2 partial coefficients in { 1 , 2 } √ 555391024 − 70937 462 ≈ 0 . 22 F = 2507812168 (Freiman’s constant) 7 / 16

  28. Motivation Markov constant Results ❚❤❡ s❡t L largest gap largest accumulation point 5 √ 221 13 1 √ √ 1 1 1571 8 0 F 1 1 √ √ √ 13 12 3 discrete part 5 continuous part (Hall’s ray) at least one partial coefficient larger than 2 partial coefficients in { 1 , 2 } explored jungle √ 555391024 − 70937 462 ≈ 0 . 22 F = 2507812168 (Freiman’s constant) 7 / 16

  29. Motivation Markov constant Results ❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S ( α ) ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ m 2 � k � � � S ( α ) = set of all accumulation points of m − α : k , m ∈ Z • S ( α ) is a closed set, closed under multiplication by squares 8 / 16

Recommend

More recommend