On the regular sum-free sets

TL;DR

This paper explores the structure of sum-free sets of natural numbers linked to specific zero-one sequences, demonstrating their regularity and automaticity, and expanding understanding of their combinatorial properties.

Contribution

It establishes that sum-free sets associated with certain zero-one sequences are 2-regular and proves their automaticity in specific cases, connecting combinatorics and automata theory.

Findings

Sum-free sets linked to Cantor-like sequences are 2-regular.

Certain sum-free sets are proven to be automatic.

The study extends the understanding of the structure of sum-free sets.

Abstract

Cameron introduced a bijection between the set of sum-free sets and the set of all zero-one sequences. In this paper, we study the sum-free sets of natural numbers corresponding to certain zero-one sequences which contain the Cantor-like sequences and some substitution sequences, etc. Those sum- free sets considered as integer sequences are 2-regular. We also prove that sequences corresponding to certain sum-free sets are automatic.