Rarefied Thue-Morse Sums Via Automata Theory and Logic

Jeffrey Shallit

Rarefied Thue-Morse Sums Via Automata Theory and Logic

Number Theory 2023-02-23 v2 Formal Languages and Automata Theory

Authors: Jeffrey Shallit

Abstract

Let $t(n)$ denote the number of $1$ -bits in the base- $2$ representation of $n$ , taken modulo $2$ . We show how to prove the classic conjecture of Leo Moser, on the rarefied sum $\sum_{0\leq i<n} (-1)^{t(3i)}$ , using tools from automata theory and logic. The same technique can be used to prove results about analogous sums.

Keywords

Thue-Morse sequence rarefied sums automata theory logic Leo Moser conjecture base-2 representation

Cite

@article{arxiv.2302.09436,
  title  = {Rarefied Thue-Morse Sums Via Automata Theory and Logic},
  author = {Jeffrey Shallit},
  journal= {arXiv preprint arXiv:2302.09436},
  year   = {2023}
}

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