Thomas Stoll: Newman's phenomenon for generalized Thue-Morse sequences

This is a joint work with Michael Drmota.

Let $ t_k=(-1)^{s(k)}$ be the classical Thue-Morse sequence with $ s(k)$ denoting the sum of the digits in the binary expansion of $ k$. A well-known result of Newman [2] says that $ t_0+t_3+t_6+\cdots+ t_{3n}>0$ for all $ n\geq 1$. Drmota and Skalba [1] showed that the arithmetic progression of indices $ 0,3,6,\dots,3n$ can be replaced by $ 0,3\kappa,6\kappa,\dots,3\kappa n$ with any fixed $ \kappa\geq 1$; positivity was shown to hold for all but finitely many $ n$. This talk is concerned about generalizations. Let $ g\geq 2$ be any given base, $ \omega_a=\exp(2\pi \imath/a)$ for $ a\geq 2$ and $ t^{(a,g)}_k=\omega_a^{s_g(k)}$ be the generalized Thue-Morse sequence, where $ s_g(k)$ denotes the sum of digits in the $ g$-ary digit expansion of $ k$. We first observe trivial Newman-like phenomena whenever $ a\vert g-1$. In the main part of the talk we show that the original case $ a=2$ inherits Newman-like phenomena for every even $ g\geq 2$ and large classes of arithmetic progressions of indices. This, in particular, also extends a result by Dumont to the general $ g$-case.


Michael Drmota and Mariusz Ska\lba.
Rarified sums of the Thue-Morse sequence.
Trans. Amer. Math. Soc., 352(2):609-642, 2000.

Donald J. Newman.
On the number of binary digits in a multiple of three.
Proc. Amer. Math. Soc., 21:719-721, 1969.

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