January  2021, 41(1): 297-327. doi: 10.3934/dcds.2020363

$ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization

1. 

LAMFA, UMR 7352 CNRS, University of Picardie, 33 rue Saint Leu, 80039 Amiens, France

2. 

School of Mathematics and Statistics, Central China Normal University 430079 Wuhan, China

3. 

Centre for Mathematical Sciences, Lund University, Box 118, 221 00 LUND, Sweden

4. 

Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, 200438 Shanghai, China

Received  October 2019 Revised  August 2020 Published  October 2020

Given an integer $ q\ge 2 $ and a real number $ c\in [0,1) $, consider the generalized Thue-Morse sequence $ (t_n^{(q;c)})_{n\ge 0} $ defined by $ t_n^{(q;c)} = e^{2\pi i c s_q(n)} $, where $ s_q(n) $ is the sum of digits of the $ q $-expansion of $ n $. We prove that the $ L^\infty $-norm of the trigonometric polynomials $ \sigma_{N}^{(q;c)} (x) : = \sum_{n = 0}^{N-1} t_n^{(q;c)} e^{2\pi i n x} $, behaves like $ N^{\gamma(q;c)} $, where $ \gamma(q;c) $ is equal to the dynamical maximal value of $ \log_q \left|\frac{\sin q\pi (x+c)}{\sin \pi (x+c)}\right| $ relative to the dynamics $ x \mapsto qx \mod 1 $ and that the maximum value is attained by a $ q $-Sturmian measure. Numerical values of $ \gamma(q;c) $ can be computed.

Citation: Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363
References:
[1]

C. Aistleitner, R. Hofer and G. Larcher, On evil Kronecker sequences and lacunary trigonometric products, Ann. Inst. Fourier (Grenoble), 67 (2017), 637–687. doi: 10.5802/aif.3094.  Google Scholar

[2]

V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard, Entrance time functions for flat spot maps, Nonlinearity, 23 (2010), 1477–1494. doi: 10.1088/0951-7715/23/6/011.  Google Scholar

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V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, The flat spot standard family: Variation of the entrance time median, Dyn. Syst., 27 (2012), 29–43. doi: 10.1080/14689367.2011.625553.  Google Scholar

[4]

V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, Sturmian maximizing measures for the piecewise-linear cosine family, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 285–302. doi: 10.1007/s00574-012-0013-3.  Google Scholar

[5]

J. Bochi, Ergodic opitimization of Birkhoff averages and Lyapunov exponents, Proc. Int. Cong. Math. 2018 Rio de Janeiro, 3 (2018), 1825–1846.  Google Scholar

[6]

T. Bousch, Le poisson n'a pas d'arêtes, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 489–508. doi: 10.1016/S0246-0203(00)00132-1.  Google Scholar

[7]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287–311. doi: 10.1016/S0012-9593(00)01062-4.  Google Scholar

[8]

T. Bousch and O. Jenkinson, Cohomology classes of dynamically non-negative $C^k$ functions, Invent. Math., 148 (2002), 207–217. doi: 10.1007/s002220100194.  Google Scholar

[9]

C. Boyd, On the structure of the family of Cherry fields on the torus, Ergodic Theory Dynam. Systems, 5 (1985), 27–46. doi: 10.1017/S014338570000273X.  Google Scholar

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S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc., 115 (1994), 451–481. doi: 10.1017/S0305004100072236.  Google Scholar

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G. Contreras, A. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1379–1409. doi: 10.1017/S0143385701001663.  Google Scholar

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G. Contreras, Ground states are generically a periodic orbit, Invent. Math., 205 (2016), 383–412. doi: 10.1007/s00222-015-0638-0.  Google Scholar

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J. P. Conze and Yves Guivarc'h, Croissance des sommes ergodiques et principe variationnel, Unpublished preprint. Google Scholar

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C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d'entiers, Ann. Inst. Fourier (Grenoble), 55 (2005), 2423–2474. doi: 10.5802/aif.2166.  Google Scholar

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A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory Dynam. Systems, 39 (2019), 1275–1289. doi: 10.1017/etds.2017.81.  Google Scholar

[16]

A. Fan, J. Schmeling and W. Shen, Multifractal analysis of generalized Thue-Morse polynomials, In preparation. Google Scholar

[17]

A. Fan and J. Konieczny, On uniformity of $q$-multiplicative sequences, Bull. Lond. Math. Soc., 51 (2019), 466–488. doi: 10.1112/blms.12245.  Google Scholar

[18]

E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres, Acta Arith., 77 (1996), 339–351. doi: 10.4064/aa-77-4-339-351.  Google Scholar

[19]

E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, Math. Ann., 305 (1996), 571–599. doi: 10.1007/BF01444238.  Google Scholar

[20]

A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259–265. doi: 10.4064/aa-13-3-259-265.  Google Scholar

[21]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233.  Google Scholar

[22]

O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems, 39 (2019), 2593–2618. doi: 10.1017/etds.2017.142.  Google Scholar

[23]

O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197–224. doi: 10.3934/dcds.2006.15.197.  Google Scholar

[24]

O. Jenkinson, Optimization and majorization of invariant measures, Electron. Res. Announc. Amer. Math. Soc., 13 (2007), 1–12. doi: 10.1090/S1079-6762-07-00170-9.  Google Scholar

[25]

O. Jenkinson, A partial order on $\times2$-invariant measures, Math. Res. Lett., 15 (2008), 893–900. doi: 10.4310/MRL.2008.v15.n5.a6.  Google Scholar

[26]

O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl., 1 (2009), 463–483. doi: 10.1142/S179383090900035X.  Google Scholar

[27]

O. Jenkinson, R. D. Mauldin and M. Urbański, Ergodic optimization for noncompact dynamical systems, Dyn. Syst., 22 (2007), 379–388. doi: 10.1080/14689360701450543.  Google Scholar

[28]

O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures and the finiteness conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 3062–3100. doi: 10.1017/etds.2017.18.  Google Scholar

[29]

O. Jenkinson and J. Steel, Majorization of invariant measures for orientation-reversing maps, Ergodic Theory Dynam. Systems, 30 (2010), 1471–1483. doi: 10.1017/S0143385709000686.  Google Scholar

[30]

J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, Ann. Inst. Fourier (Grenoble), 69 (2019), 1897–1913. doi: 10.5802/aif.3285.  Google Scholar

[31]

K. Mahler, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions: Part two on the translation properties of a simple class of arithmetical functions, Journal of Mathematics and Physics, 6 (1927), 158–163. doi: 10.1002/sapm192761158.  Google Scholar

[32]

C. Mauduit and J. Rivat, La somme des chiffres des carrés, Acta Math., 203 (2009), 107–148. doi: 10.1007/s11511-009-0040-0.  Google Scholar

[33]

C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. (2), 171 (2010), 1591–1646. doi: 10.4007/annals.2010.171.1591.  Google Scholar

[34]

C. Mauduit, J. Rivat and A. Sárközy, On the digits of sumsets, Canad. J. Math., 69 (2017), 595–612. doi: 10.4153/CJM-2016-007-2.  Google Scholar

[35]

V. A. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268–282.  Google Scholar

[36]

M. Queffélec, Questions around the Thue-Morse sequence, Unif. Distrib. Theory, 13 (2018), 1–25. doi: 10.1515/udt-2018-0001.  Google Scholar

[37]

J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419–428. doi: 10.1088/0951-7715/2/3/003.  Google Scholar

[38]

Y. Zhang, K. Yin and W. Wu, A rigorous computer aided estimation for Gelfond exponent of weighted Thue-Morse sequences, arXiv: 1806.08329v2. Google Scholar

show all references

References:
[1]

C. Aistleitner, R. Hofer and G. Larcher, On evil Kronecker sequences and lacunary trigonometric products, Ann. Inst. Fourier (Grenoble), 67 (2017), 637–687. doi: 10.5802/aif.3094.  Google Scholar

[2]

V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard, Entrance time functions for flat spot maps, Nonlinearity, 23 (2010), 1477–1494. doi: 10.1088/0951-7715/23/6/011.  Google Scholar

[3]

V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, The flat spot standard family: Variation of the entrance time median, Dyn. Syst., 27 (2012), 29–43. doi: 10.1080/14689367.2011.625553.  Google Scholar

[4]

V. Anagnostopoulou, K. Díaz-Ordaz, O. Jenkinson and C. Richard,, Sturmian maximizing measures for the piecewise-linear cosine family, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 285–302. doi: 10.1007/s00574-012-0013-3.  Google Scholar

[5]

J. Bochi, Ergodic opitimization of Birkhoff averages and Lyapunov exponents, Proc. Int. Cong. Math. 2018 Rio de Janeiro, 3 (2018), 1825–1846.  Google Scholar

[6]

T. Bousch, Le poisson n'a pas d'arêtes, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 489–508. doi: 10.1016/S0246-0203(00)00132-1.  Google Scholar

[7]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287–311. doi: 10.1016/S0012-9593(00)01062-4.  Google Scholar

[8]

T. Bousch and O. Jenkinson, Cohomology classes of dynamically non-negative $C^k$ functions, Invent. Math., 148 (2002), 207–217. doi: 10.1007/s002220100194.  Google Scholar

[9]

C. Boyd, On the structure of the family of Cherry fields on the torus, Ergodic Theory Dynam. Systems, 5 (1985), 27–46. doi: 10.1017/S014338570000273X.  Google Scholar

[10]

S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc., 115 (1994), 451–481. doi: 10.1017/S0305004100072236.  Google Scholar

[11]

G. Contreras, A. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1379–1409. doi: 10.1017/S0143385701001663.  Google Scholar

[12]

G. Contreras, Ground states are generically a periodic orbit, Invent. Math., 205 (2016), 383–412. doi: 10.1007/s00222-015-0638-0.  Google Scholar

[13]

J. P. Conze and Yves Guivarc'h, Croissance des sommes ergodiques et principe variationnel, Unpublished preprint. Google Scholar

[14]

C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d'entiers, Ann. Inst. Fourier (Grenoble), 55 (2005), 2423–2474. doi: 10.5802/aif.2166.  Google Scholar

[15]

A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory Dynam. Systems, 39 (2019), 1275–1289. doi: 10.1017/etds.2017.81.  Google Scholar

[16]

A. Fan, J. Schmeling and W. Shen, Multifractal analysis of generalized Thue-Morse polynomials, In preparation. Google Scholar

[17]

A. Fan and J. Konieczny, On uniformity of $q$-multiplicative sequences, Bull. Lond. Math. Soc., 51 (2019), 466–488. doi: 10.1112/blms.12245.  Google Scholar

[18]

E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres, Acta Arith., 77 (1996), 339–351. doi: 10.4064/aa-77-4-339-351.  Google Scholar

[19]

E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, Math. Ann., 305 (1996), 571–599. doi: 10.1007/BF01444238.  Google Scholar

[20]

A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259–265. doi: 10.4064/aa-13-3-259-265.  Google Scholar

[21]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233.  Google Scholar

[22]

O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems, 39 (2019), 2593–2618. doi: 10.1017/etds.2017.142.  Google Scholar

[23]

O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197–224. doi: 10.3934/dcds.2006.15.197.  Google Scholar

[24]

O. Jenkinson, Optimization and majorization of invariant measures, Electron. Res. Announc. Amer. Math. Soc., 13 (2007), 1–12. doi: 10.1090/S1079-6762-07-00170-9.  Google Scholar

[25]

O. Jenkinson, A partial order on $\times2$-invariant measures, Math. Res. Lett., 15 (2008), 893–900. doi: 10.4310/MRL.2008.v15.n5.a6.  Google Scholar

[26]

O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl., 1 (2009), 463–483. doi: 10.1142/S179383090900035X.  Google Scholar

[27]

O. Jenkinson, R. D. Mauldin and M. Urbański, Ergodic optimization for noncompact dynamical systems, Dyn. Syst., 22 (2007), 379–388. doi: 10.1080/14689360701450543.  Google Scholar

[28]

O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures and the finiteness conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 3062–3100. doi: 10.1017/etds.2017.18.  Google Scholar

[29]

O. Jenkinson and J. Steel, Majorization of invariant measures for orientation-reversing maps, Ergodic Theory Dynam. Systems, 30 (2010), 1471–1483. doi: 10.1017/S0143385709000686.  Google Scholar

[30]

J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, Ann. Inst. Fourier (Grenoble), 69 (2019), 1897–1913. doi: 10.5802/aif.3285.  Google Scholar

[31]

K. Mahler, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions: Part two on the translation properties of a simple class of arithmetical functions, Journal of Mathematics and Physics, 6 (1927), 158–163. doi: 10.1002/sapm192761158.  Google Scholar

[32]

C. Mauduit and J. Rivat, La somme des chiffres des carrés, Acta Math., 203 (2009), 107–148. doi: 10.1007/s11511-009-0040-0.  Google Scholar

[33]

C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. (2), 171 (2010), 1591–1646. doi: 10.4007/annals.2010.171.1591.  Google Scholar

[34]

C. Mauduit, J. Rivat and A. Sárközy, On the digits of sumsets, Canad. J. Math., 69 (2017), 595–612. doi: 10.4153/CJM-2016-007-2.  Google Scholar

[35]

V. A. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268–282.  Google Scholar

[36]

M. Queffélec, Questions around the Thue-Morse sequence, Unif. Distrib. Theory, 13 (2018), 1–25. doi: 10.1515/udt-2018-0001.  Google Scholar

[37]

J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419–428. doi: 10.1088/0951-7715/2/3/003.  Google Scholar

[38]

Y. Zhang, K. Yin and W. Wu, A rigorous computer aided estimation for Gelfond exponent of weighted Thue-Morse sequences, arXiv: 1806.08329v2. Google Scholar

Figure 1.  The graphs of $ f_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $
Figure 2.  The graphs of $ \gamma(2;c) $. Only the cycles of order $ \le 13 $ are used to plot the graph. To fill in the gaps in the graph, we have to use other cycles (there are infinitely many)
Figure 3.  The graphs of $ \log|2\sin \pi (x-b)| $ on the intervals $ [0,1] $ and $ [b,b+1] $ with $ b = 1/3 $
Figure 4.  The graphs of $ f'_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $
Figure 5.  The branch $ T|_{C_\lambda} $
Figure 6.  The graphs of $ e_0 $ and $ e_{1/4} $
Table 1.  Values of $ \beta(c) $ and $ \gamma(c) $ for specific $ c $'s
$c$ $\beta(c)$ $ \gamma(c)$ $ c $ $ \beta(c)$ $\gamma(c)$
$1/2$ $\log(\sqrt{3})$ $ \log 3/\log 4$ $7/18$ $0.51079$ $ 0.73691 $
$1/3$ $0.52227$ $0.75347$ $4/19$ $0.51949$ $0.74947$
$1/4$ $0.51586$ $0.74423$ $5/19$ $0.51719$ $0.74615$
$1/5$ $0.52201$ $0.75310$ $6/19$ $0.51830$ $0.74775$
$2/5$ $0.51217$ $0.73890$ $7/19$ $0.51701$ $0.74589$
$2/7$ $0.51354$ $0.74088$ $8/19$ $0.51252$ $0.73941$
$3/7$ $0.51515$ $0.74321$ $9/19$ $0.54474$ $0.78589$
$3/8$ $0.51406$ $0.74163$ $7/20$ $0.52195$ $0.75302$
$2/9$ $0.51848$ $0.74802$ $9/20$ $0.53272$ $0.76855$
$4/9$ $0.52879$ $0.76288$ $4/21$ $0.52489$ $0.75725$
$3/10$ $0.51184$ $0.73843$ $5/21$ $0.51576$ $0.74408$
$2/11$ $0.52852$ $0.76250$ $8/21$ **
$3/11$ $0.51655$ $0.74523$ $5/22$ $0.51802$ $0.74735$
$4/11$ $0.51875$ $0.74840$ $7/22$ $0.51910$ $0.74891$
$5/11$ $0.53562$ $0.77273$ $9/22$ $0.51196$ $0.73860$
$5/12$ $0.51185$ $0.73844$ $5/23$ $0.51857$ $0.74814$
$3/13$ $0.51748$ $0.74657$ $6/23$ $0.51714$ $0.74608$
$4/13$ $0.51496$ $0.74293$ $7/23$ $0.51329$ $0.74052$
$5/13$ $0.49827$ $0.71885$ $8/23$ $0.52222$ $0.75340$
$6/13$ $0.53952$ $0.77837$ $9/23$ $0.51124$ $0.73756$
$3/14$ $0.51844$ $0.74795$ $10/23$ $0.52092$ $0.75153$
$5/14$ $0.52061$ $0.75108$ $11/23$ $0.54619$ $0.78799$
$7/15$ $0.54197$ $0.78190$ $5/24$ $0.52015$ $0.75042$
$7/16$ $0.52326$ $0.75491$ $7/24$ $0.51179$ $0.73836$
$3/17$ $0.53203$ $0.76756$ $11/24$ $0.53782$ $0.77591$
$4/17$ $0.51651$ $0.74516$ $6/25$ $0.51517$ $0.74324$
$5/17$ $0.51191$ $0.73853$ $7/25$ $0.515168$ $0.74323$
$6/17$ $0.52148$ $0.75234$ $8/25$ $0.51966$ $0.74971$
$7/17$ $0.51167$ $0.73818$ $9/25$ $0.51987$ $0.75001$
$8/17$ $0.54360$ $0.78425$ $11/25$ $0.52534$ $0.75789$
$5/18$ $0.51567$ $0.74396$ $12/25$ $0.54667$ $0.78868$
  ** We don't compute $\beta(c)$ and $\gamma(c)$ if the parameter $c$ doesn't belong to any of the intervals in Table 2.
$c$ $\beta(c)$ $ \gamma(c)$ $ c $ $ \beta(c)$ $\gamma(c)$
$1/2$ $\log(\sqrt{3})$ $ \log 3/\log 4$ $7/18$ $0.51079$ $ 0.73691 $
$1/3$ $0.52227$ $0.75347$ $4/19$ $0.51949$ $0.74947$
$1/4$ $0.51586$ $0.74423$ $5/19$ $0.51719$ $0.74615$
$1/5$ $0.52201$ $0.75310$ $6/19$ $0.51830$ $0.74775$
$2/5$ $0.51217$ $0.73890$ $7/19$ $0.51701$ $0.74589$
$2/7$ $0.51354$ $0.74088$ $8/19$ $0.51252$ $0.73941$
$3/7$ $0.51515$ $0.74321$ $9/19$ $0.54474$ $0.78589$
$3/8$ $0.51406$ $0.74163$ $7/20$ $0.52195$ $0.75302$
$2/9$ $0.51848$ $0.74802$ $9/20$ $0.53272$ $0.76855$
$4/9$ $0.52879$ $0.76288$ $4/21$ $0.52489$ $0.75725$
$3/10$ $0.51184$ $0.73843$ $5/21$ $0.51576$ $0.74408$
$2/11$ $0.52852$ $0.76250$ $8/21$ **
$3/11$ $0.51655$ $0.74523$ $5/22$ $0.51802$ $0.74735$
$4/11$ $0.51875$ $0.74840$ $7/22$ $0.51910$ $0.74891$
$5/11$ $0.53562$ $0.77273$ $9/22$ $0.51196$ $0.73860$
$5/12$ $0.51185$ $0.73844$ $5/23$ $0.51857$ $0.74814$
$3/13$ $0.51748$ $0.74657$ $6/23$ $0.51714$ $0.74608$
$4/13$ $0.51496$ $0.74293$ $7/23$ $0.51329$ $0.74052$
$5/13$ $0.49827$ $0.71885$ $8/23$ $0.52222$ $0.75340$
$6/13$ $0.53952$ $0.77837$ $9/23$ $0.51124$ $0.73756$
$3/14$ $0.51844$ $0.74795$ $10/23$ $0.52092$ $0.75153$
$5/14$ $0.52061$ $0.75108$ $11/23$ $0.54619$ $0.78799$
$7/15$ $0.54197$ $0.78190$ $5/24$ $0.52015$ $0.75042$
$7/16$ $0.52326$ $0.75491$ $7/24$ $0.51179$ $0.73836$
$3/17$ $0.53203$ $0.76756$ $11/24$ $0.53782$ $0.77591$
$4/17$ $0.51651$ $0.74516$ $6/25$ $0.51517$ $0.74324$
$5/17$ $0.51191$ $0.73853$ $7/25$ $0.515168$ $0.74323$
$6/17$ $0.52148$ $0.75234$ $8/25$ $0.51966$ $0.74971$
$7/17$ $0.51167$ $0.73818$ $9/25$ $0.51987$ $0.75001$
$8/17$ $0.54360$ $0.78425$ $11/25$ $0.52534$ $0.75789$
$5/18$ $0.51567$ $0.74396$ $12/25$ $0.54667$ $0.78868$
  ** We don't compute $\beta(c)$ and $\gamma(c)$ if the parameter $c$ doesn't belong to any of the intervals in Table 2.
Table 2.  Valid intervals $[c_*, c^*]$
Period $s_{\max}-\frac{1}{2}$ $s_{\min}$ $ [c_*,c^*] $
$1$ $-1/2$ $0$ $[0.000000000000000,0.175160000000000]$
$2$ $1/6$ $1/3$ $[0.428133329021334,0.571866670978666]$
$ 3 $ $1/14 $ $ 1/7 $ $ [0.619203577131485,0.697872156658965] $
$ 3 $ $5/14 $ $ 3/7 $ $ [0.302127843341035,0.380796422868515] $
$ 4 $ $1/30 $ $ 1/15 $ $ [0.709633870795466,0.755421357085333] $
$ 4 $ $13/30 $ $ 7/15 $ $ [0.244578642914667,0.290366129204534] $
$ 5 $ $1/62 $ $ 1/31 $ $ [0.758710839860046,0.785842721390351] $
$ 5 $ $29/62$ $15/31 $ $ [0.214157278609649,0.241289160139954] $
$ 5 $ $9/62$ $ 5/31$ $ [0.586141644350735,0.612800854796395] $
$ 5 $ $21/62$ $11/31$ $[0.387199145203605,0.413858355649265] $
$ 6 $ $1/126$ $1/63$ $[0.786809543609523,0.802555581755556] $
$ 6 $ $61/126$ $31/63$ $[0.197444418244444,0.213190456390477] $
$ 7 $ $1/254$ $1/127$ $[0.803225220690394,0.812352783425512] $
$ 7 $ $125/254$ $63/127$ $[0.187647216574488,0.196774779309606] $
$ 7 $ $17/254$ $9/127$ $[0.699811031164904,0.708527570112261] $
$ 7 $ $109/254$ $55/127$ $[0.291472429887739,0.300188968835096] $
$ 7 $ $41/254$ $21/127$ $[0.576825192903727,0.585555905085145] $
$ 7 $ $85/254$ $43/127$ $[0.414444094914855,0.423174807096273] $
$ 8 $ $1/510$ $1/255$ $[0.812634013261438,0.817780420556863] $
$ 8 $ $253/510$ $127/255$ $[0.182219579443137,0.187365986738562] $
$ 8 $ $73/510$ $37/255$ $[0.613186931037909,0.617835298917647] $
$ 8 $ $181/510$ $91/255$ $[0.382164701082353,0.386813068962091] $
$ 9 $ $1/1022$ $1/511$ $[0.818062650175864,0.820724099383431] $
$ 9 $ $509/1022$ $255/511$ $[0.179275900616569,0.181937349824136] $
$ 9 $ $33/1022$ $17/511$ $[0.755812148539074,0.758473597746640] $
$ 9 $ $477/1022$ $239/511$ $[0.241526402253360,0.244187851460926] $
$ 9 $ $169/1022$ $85/511$ $[0.573835612305023,0.576497061512589] $
$ 9 $ $341/1022$ $171/511$ $[0.423502938487411,0.426164387694977] $
$10 $ $1/2046$ $1/1023$ $[0.821196509738417,0.822528540248941] $
$10 $ $1021/2046$ $511/1023$ $[0.177471459751059,0.178803490261583] $
$10 $ $145/2046$ $73/1023$ $[0.698241698854594,0.699698607225480] $
$10 $ $877/2046$ $439/1023$ $[0.300301392774520,0.301758301145406] $
$11 $ $1/4094$ $1/2047$ $[0.822722890076930,0.823555816343776] $
$11 $ $2045/4094$ $1023/2047$ $[0.176444183656224,0.177277109923070] $
$11 $ $65/4094$ $33/2047$ $[0.786058868717432,0.786683563417567] $
$11 $ $1981/4094$ $991/2047$ $[0.213316436582433,0.213941131282568] $
$11 $ $273/4094$ $137/2047$ $[0.708807402099743,0.709432096799878] $
$11 $ $1773/4094$ $887/2047$ $[0.290567903200122,0.291192597900257] $
$11 $ $585/4094$ $293/2047$ $[0.618230337528651,0.619009737929774] $
$11 $ $1461/4094$ $731/2047$ $[0.380990262070226,0.381769662471349] $
$11 $ $681/4094$ $341/2047$ $[0.572917073341487,0.573541768041622] $
$11 $ $1365/4094$ $683/2047$ $[0.426458231958378,0.427082926658513] $
$12 $ $1/8190$ $1/4095$ $[0.823705054848802,0.824017478548726] $
$12 $ $4093/8190$ $2047/4095$ $[0.175982521451274,0.176294945151198] $
$12 $ $1321/8190$ $661/4095$ $[0.585676663495414,0.585989087195338] $
$12 $ $2773/8190$ $1387/4095$ $[0.414010912804662,0.414323336504586] $
$13 $ $1/16382$ $1/8191$ $[0.824099377662201,0.824366011773877] $
$13 $ $8189/16382$ $4095/8191$ $[0.175633988226123,0.175900622337799] $
$13 $ $129/16382$ $65/8191$ $[0.802834232937408,0.803074203637915] $
$13 $ $8061/16382$ $4031/8191$ $[0.196925796362085,0.197165767062592] $
$13 $ $545/16382$ $273/8191$ $[0.755457525487725,0.755686164238488] $
$13 $ $7645/16382$ $3823/8191$ $[0.244313835761512,0.244542474512275] $
$13 $ $1169/16382$ $585/8191$ $[0.697932644443065,0.698161283193827] $
$13 $ $7021/16382$ $3511/8191$ $[0.301838716806173,0.302067355556935] $
$13 $ $2377/16382$ $1189/8191$ $[0.612842893451498,0.613081331005864] $
$13 $ $5813/16382$ $2907/8191$ $[0.386918668994136,0.387157106548502] $
$13 $ $2729/16382$ $1365/8191$ $[0.572640180643138,0.572864153296945] $
$13 $ $5461/16382$ $2731/8191$ $[0.427135846703055,0.427359819356862] $
Period $s_{\max}-\frac{1}{2}$ $s_{\min}$ $ [c_*,c^*] $
$1$ $-1/2$ $0$ $[0.000000000000000,0.175160000000000]$
$2$ $1/6$ $1/3$ $[0.428133329021334,0.571866670978666]$
$ 3 $ $1/14 $ $ 1/7 $ $ [0.619203577131485,0.697872156658965] $
$ 3 $ $5/14 $ $ 3/7 $ $ [0.302127843341035,0.380796422868515] $
$ 4 $ $1/30 $ $ 1/15 $ $ [0.709633870795466,0.755421357085333] $
$ 4 $ $13/30 $ $ 7/15 $ $ [0.244578642914667,0.290366129204534] $
$ 5 $ $1/62 $ $ 1/31 $ $ [0.758710839860046,0.785842721390351] $
$ 5 $ $29/62$ $15/31 $ $ [0.214157278609649,0.241289160139954] $
$ 5 $ $9/62$ $ 5/31$ $ [0.586141644350735,0.612800854796395] $
$ 5 $ $21/62$ $11/31$ $[0.387199145203605,0.413858355649265] $
$ 6 $ $1/126$ $1/63$ $[0.786809543609523,0.802555581755556] $
$ 6 $ $61/126$ $31/63$ $[0.197444418244444,0.213190456390477] $
$ 7 $ $1/254$ $1/127$ $[0.803225220690394,0.812352783425512] $
$ 7 $ $125/254$ $63/127$ $[0.187647216574488,0.196774779309606] $
$ 7 $ $17/254$ $9/127$ $[0.699811031164904,0.708527570112261] $
$ 7 $ $109/254$ $55/127$ $[0.291472429887739,0.300188968835096] $
$ 7 $ $41/254$ $21/127$ $[0.576825192903727,0.585555905085145] $
$ 7 $ $85/254$ $43/127$ $[0.414444094914855,0.423174807096273] $
$ 8 $ $1/510$ $1/255$ $[0.812634013261438,0.817780420556863] $
$ 8 $ $253/510$ $127/255$ $[0.182219579443137,0.187365986738562] $
$ 8 $ $73/510$ $37/255$ $[0.613186931037909,0.617835298917647] $
$ 8 $ $181/510$ $91/255$ $[0.382164701082353,0.386813068962091] $
$ 9 $ $1/1022$ $1/511$ $[0.818062650175864,0.820724099383431] $
$ 9 $ $509/1022$ $255/511$ $[0.179275900616569,0.181937349824136] $
$ 9 $ $33/1022$ $17/511$ $[0.755812148539074,0.758473597746640] $
$ 9 $ $477/1022$ $239/511$ $[0.241526402253360,0.244187851460926] $
$ 9 $ $169/1022$ $85/511$ $[0.573835612305023,0.576497061512589] $
$ 9 $ $341/1022$ $171/511$ $[0.423502938487411,0.426164387694977] $
$10 $ $1/2046$ $1/1023$ $[0.821196509738417,0.822528540248941] $
$10 $ $1021/2046$ $511/1023$ $[0.177471459751059,0.178803490261583] $
$10 $ $145/2046$ $73/1023$ $[0.698241698854594,0.699698607225480] $
$10 $ $877/2046$ $439/1023$ $[0.300301392774520,0.301758301145406] $
$11 $ $1/4094$ $1/2047$ $[0.822722890076930,0.823555816343776] $
$11 $ $2045/4094$ $1023/2047$ $[0.176444183656224,0.177277109923070] $
$11 $ $65/4094$ $33/2047$ $[0.786058868717432,0.786683563417567] $
$11 $ $1981/4094$ $991/2047$ $[0.213316436582433,0.213941131282568] $
$11 $ $273/4094$ $137/2047$ $[0.708807402099743,0.709432096799878] $
$11 $ $1773/4094$ $887/2047$ $[0.290567903200122,0.291192597900257] $
$11 $ $585/4094$ $293/2047$ $[0.618230337528651,0.619009737929774] $
$11 $ $1461/4094$ $731/2047$ $[0.380990262070226,0.381769662471349] $
$11 $ $681/4094$ $341/2047$ $[0.572917073341487,0.573541768041622] $
$11 $ $1365/4094$ $683/2047$ $[0.426458231958378,0.427082926658513] $
$12 $ $1/8190$ $1/4095$ $[0.823705054848802,0.824017478548726] $
$12 $ $4093/8190$ $2047/4095$ $[0.175982521451274,0.176294945151198] $
$12 $ $1321/8190$ $661/4095$ $[0.585676663495414,0.585989087195338] $
$12 $ $2773/8190$ $1387/4095$ $[0.414010912804662,0.414323336504586] $
$13 $ $1/16382$ $1/8191$ $[0.824099377662201,0.824366011773877] $
$13 $ $8189/16382$ $4095/8191$ $[0.175633988226123,0.175900622337799] $
$13 $ $129/16382$ $65/8191$ $[0.802834232937408,0.803074203637915] $
$13 $ $8061/16382$ $4031/8191$ $[0.196925796362085,0.197165767062592] $
$13 $ $545/16382$ $273/8191$ $[0.755457525487725,0.755686164238488] $
$13 $ $7645/16382$ $3823/8191$ $[0.244313835761512,0.244542474512275] $
$13 $ $1169/16382$ $585/8191$ $[0.697932644443065,0.698161283193827] $
$13 $ $7021/16382$ $3511/8191$ $[0.301838716806173,0.302067355556935] $
$13 $ $2377/16382$ $1189/8191$ $[0.612842893451498,0.613081331005864] $
$13 $ $5813/16382$ $2907/8191$ $[0.386918668994136,0.387157106548502] $
$13 $ $2729/16382$ $1365/8191$ $[0.572640180643138,0.572864153296945] $
$13 $ $5461/16382$ $2731/8191$ $[0.427135846703055,0.427359819356862] $
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