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.

Bibliography

1

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

2

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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