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.

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\ begin {document} $ L ^ \ infty $ \ end {document}Thue-Morse三角多项式的估计和遍历最大化

给定一个整数$ q \ ge 2 $和一个实数$ c \ in [0,1）$，请考虑广义的Thue-Morse序列$（t_n ^ {（q; c）}）_ {n \ ge 0} $由$ t_n ^ {（q; c）} = e ^ {2 \ pi ic s_q（n）} $定义，其中$ s_q（n）$是$ n的$ q $扩展数的总和$。我们证明了三角多项式$ \ sigma_ {N} ^ {（q; c）}（x）的$ L ^ \ infty $-范数：= \ sum_ {n = 0} ^ {N-1} t_n ^ {（q; c）} e ^ {2 \ pi inx} $，其行为类似于$ N ^ {\ gamma（q; c）} $，其中$ \ gamma（q; c）$等于动态最大值的$ \ log_q \ left | \ frac {\ sin q \ pi（x + c）} {\ sin \ pi（x + c）} \ right | $相对于动力学$ x \ mapsto qx \ mod 1 $，并且最大值通过$ q $ -Sturmian度量获得。可以计算$ \ gamma（q; c）$的数值。