Rarefied Thue-Morse Sums Via Automata Theory and Logic

TL;DR

This paper demonstrates how automata theory and logic can be used to prove a classic conjecture about the sum of a sequence defined by binary digit parity, extending to similar sums.

Contribution

It introduces a novel approach combining automata and logic to prove conjectures related to binary digit sums and their sums, including the Moser conjecture.

Findings

Proved Moser's conjecture on rarefied sums using automata theory.

Extended the technique to analogous sums with similar properties.

Showed the effectiveness of automata and logic in combinatorial number theory.

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.