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

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

Aihua Fan 1,2, , Jörg Schmeling 3, and  Weixiao Shen 4,

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  January 2021 Early access  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.

Keywords: Gelfond exponent, generalized Thue-Morse sequences, ergodic optimization, Sturmian condition, Sturmian measures.
Mathematics Subject Classification: Primary:11L07, 37A05, 37B10.
Citation: Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

J. P. Conze and Yves Guivarc'h, Croissance des sommes ergodiques et principe variationnel, Unpublished preprint.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[37]

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

[38]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[37]

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

[38]

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

Figure 1.  The graphs of $ f_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $
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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)
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Figure 3.  The graphs of $ \log|2\sin \pi (x-b)| $ on the intervals $ [0,1] $ and $ [b,b+1] $ with $ b = 1/3 $
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Figure 4.  The graphs of $ f'_0 $ on the interval $ [-1/q,1-1/q] $, here $ q = 6 $
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Figure 5.  The branch $ T|_{C_\lambda} $
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Figure 6.  The graphs of $ e_0 $ and $ e_{1/4} $
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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.
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$c$ $\beta(c)$ $ \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] $
Table Options

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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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