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

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  • 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:

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  • 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.
     | Show Table
    DownLoad: CSV

    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] $
     | Show Table
    DownLoad: CSV
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