Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences

TL;DR

This paper investigates the complexity of sum-free sets generated by period-k-folding and Sturmian sequences, showing they are not $b7$-regular, and explores their properties and differences.

Contribution

It introduces the period-k-folding sequences, generalizing the period-doubling sequence, and analyzes the non-regularity of their sum-free sets and Sturmian sequences.

Findings

Sum-free sets from period-doubling sequences are not $b7$-regular.

Sum-free sets from period-k-folding sequences also lack $b7$-regularity.

Studied sum-free and difference sets from Sturmian sequences starting with '11'.

Abstract

First, we show that the sum-free set generated by the period-doubling sequence is not $\kappa$-regular for any $\kappa\geq 2$. Next, we introduce a generalization of the period-doubling sequence, which we call the period-$k$-folding sequences. We show that the sum-free sets generated by the period-$k$-folding sequences also fail to be $\kappa$-regular for all $\kappa\geq 2$. Finally, we study the sum-free sets generated by Sturmian sequences that begin with `11', and their difference sequences.