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

## This is a joint work with Michael Drmota.

Let be the classical Thue-Morse sequence with denoting the sum of the digits in the binary expansion of . A well-known result of Newman [2] says that for all . Drmota and Skalba [1] showed that the arithmetic progression of indices can be replaced by with any fixed ; positivity was shown to hold for all but finitely many . This talk is concerned about generalizations. Let be any given base, for and be the generalized Thue-Morse sequence, where denotes the sum of digits in the -ary digit expansion of . We first observe trivial Newman-like phenomena whenever . In the main part of the talk we show that the original case inherits Newman-like phenomena for every even and large classes of arithmetic progressions of indices. This, in particular, also extends a result by Dumont to the general -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.