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

• 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.

Mathematics Subject Classification: Primary:11L07, 37A05, 37B10.

 Citation:

• 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.

Table 2.  Valid intervals $[c_*, c^*]$

 Period $s_{\max}-\frac{1}{2}$ $s_{\min}$ $[c_*,c^*]$ $1$ $-1/2$ $0$ $[0.000000000000000,0.175160000000000]$ $2$ $1/6$ $1/3$ $[0.428133329021334,0.571866670978666]$ $3$ $1/14$ $1/7$ $[0.619203577131485,0.697872156658965]$ $3$ $5/14$ $3/7$ $[0.302127843341035,0.380796422868515]$ $4$ $1/30$ $1/15$ $[0.709633870795466,0.755421357085333]$ $4$ $13/30$ $7/15$ $[0.244578642914667,0.290366129204534]$ $5$ $1/62$ $1/31$ $[0.758710839860046,0.785842721390351]$ $5$ $29/62$ $15/31$ $[0.214157278609649,0.241289160139954]$ $5$ $9/62$ $5/31$ $[0.586141644350735,0.612800854796395]$ $5$ $21/62$ $11/31$ $[0.387199145203605,0.413858355649265]$ $6$ $1/126$ $1/63$ $[0.786809543609523,0.802555581755556]$ $6$ $61/126$ $31/63$ $[0.197444418244444,0.213190456390477]$ $7$ $1/254$ $1/127$ $[0.803225220690394,0.812352783425512]$ $7$ $125/254$ $63/127$ $[0.187647216574488,0.196774779309606]$ $7$ $17/254$ $9/127$ $[0.699811031164904,0.708527570112261]$ $7$ $109/254$ $55/127$ $[0.291472429887739,0.300188968835096]$ $7$ $41/254$ $21/127$ $[0.576825192903727,0.585555905085145]$ $7$ $85/254$ $43/127$ $[0.414444094914855,0.423174807096273]$ $8$ $1/510$ $1/255$ $[0.812634013261438,0.817780420556863]$ $8$ $253/510$ $127/255$ $[0.182219579443137,0.187365986738562]$ $8$ $73/510$ $37/255$ $[0.613186931037909,0.617835298917647]$ $8$ $181/510$ $91/255$ $[0.382164701082353,0.386813068962091]$ $9$ $1/1022$ $1/511$ $[0.818062650175864,0.820724099383431]$ $9$ $509/1022$ $255/511$ $[0.179275900616569,0.181937349824136]$ $9$ $33/1022$ $17/511$ $[0.755812148539074,0.758473597746640]$ $9$ $477/1022$ $239/511$ $[0.241526402253360,0.244187851460926]$ $9$ $169/1022$ $85/511$ $[0.573835612305023,0.576497061512589]$ $9$ $341/1022$ $171/511$ $[0.423502938487411,0.426164387694977]$ $10$ $1/2046$ $1/1023$ $[0.821196509738417,0.822528540248941]$ $10$ $1021/2046$ $511/1023$ $[0.177471459751059,0.178803490261583]$ $10$ $145/2046$ $73/1023$ $[0.698241698854594,0.699698607225480]$ $10$ $877/2046$ $439/1023$ $[0.300301392774520,0.301758301145406]$ $11$ $1/4094$ $1/2047$ $[0.822722890076930,0.823555816343776]$ $11$ $2045/4094$ $1023/2047$ $[0.176444183656224,0.177277109923070]$ $11$ $65/4094$ $33/2047$ $[0.786058868717432,0.786683563417567]$ $11$ $1981/4094$ $991/2047$ $[0.213316436582433,0.213941131282568]$ $11$ $273/4094$ $137/2047$ $[0.708807402099743,0.709432096799878]$ $11$ $1773/4094$ $887/2047$ $[0.290567903200122,0.291192597900257]$ $11$ $585/4094$ $293/2047$ $[0.618230337528651,0.619009737929774]$ $11$ $1461/4094$ $731/2047$ $[0.380990262070226,0.381769662471349]$ $11$ $681/4094$ $341/2047$ $[0.572917073341487,0.573541768041622]$ $11$ $1365/4094$ $683/2047$ $[0.426458231958378,0.427082926658513]$ $12$ $1/8190$ $1/4095$ $[0.823705054848802,0.824017478548726]$ $12$ $4093/8190$ $2047/4095$ $[0.175982521451274,0.176294945151198]$ $12$ $1321/8190$ $661/4095$ $[0.585676663495414,0.585989087195338]$ $12$ $2773/8190$ $1387/4095$ $[0.414010912804662,0.414323336504586]$ $13$ $1/16382$ $1/8191$ $[0.824099377662201,0.824366011773877]$ $13$ $8189/16382$ $4095/8191$ $[0.175633988226123,0.175900622337799]$ $13$ $129/16382$ $65/8191$ $[0.802834232937408,0.803074203637915]$ $13$ $8061/16382$ $4031/8191$ $[0.196925796362085,0.197165767062592]$ $13$ $545/16382$ $273/8191$ $[0.755457525487725,0.755686164238488]$ $13$ $7645/16382$ $3823/8191$ $[0.244313835761512,0.244542474512275]$ $13$ $1169/16382$ $585/8191$ $[0.697932644443065,0.698161283193827]$ $13$ $7021/16382$ $3511/8191$ $[0.301838716806173,0.302067355556935]$ $13$ $2377/16382$ $1189/8191$ $[0.612842893451498,0.613081331005864]$ $13$ $5813/16382$ $2907/8191$ $[0.386918668994136,0.387157106548502]$ $13$ $2729/16382$ $1365/8191$ $[0.572640180643138,0.572864153296945]$ $13$ $5461/16382$ $2731/8191$ $[0.427135846703055,0.427359819356862]$
•  [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.

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