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Register a new userScaling properties of the Thue--Morse measure
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Authors
Michael Baake
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The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.
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We revisit the well-known and much studied Riesz product representation of the Thue-Morse diffraction measure, which is also the maximal spectral measure for the corresponding dynamical spectrum in the complement of the pure point part. The known scaling relations are summarised, and some new findings are explained.
Given an integer $qge 2$ and a real number $cin [0,1)$, consider the generalized Thue-Morse sequence $(t_n^{(q;c)})_{nge 0}$ defined by $t_n^{(q;c)} = e^{2pi i c S_q(n)}$, 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 $sigma_{N}^{(q;c)} (x) := sum_{n=0}^{N-1} t_n^{(q;c)} e^{2pi i n x}$, behaves like $N^{gamma(q;c)}$, where $gamma(q;c)$ is equal to the dynamical maximal value of $log_q left|frac{sin qpi (x+c)}{sin pi (x+c)}right|$ 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.
In this paper, we will provide a mathematically rigorous computer aided estimation for the exact values and robustness for Gelfond exponent of weighted Thue-Morse sequences. This result improves previous discussions on Gelfond exponent by Gelfond, Devenport, Mauduit, Rivat, S{a}rk{o}zy and Fan et. al.
We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.
Let $b ge 2$ and $ell ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ prod_{h = 0}^ell , (1 - b^{-2^h}), $$ of denominator $b^{2^{ell+1} - 1}$, do not exceed $exp(K (log b)^2 sqrt{ell} 2^{ell/2})$.
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