Dynamics of $\mathscr{B}$-free systems generated by Behrend sets. I

TL;DR

This paper investigates the complexity and dynamical properties of $\

Contribution

It characterizes the complexity growth of $\\mathscr{B}$-free subshifts generated by Behrend sets and links their transitivity to coprimality conditions.

Findings

Complexity of $\\mathscr{B}$-free subshifts can grow subexponentially.

Transitivity of $\\mathscr{B}$-admissible subshifts occurs only for coprime sets.

Characterization of subshifts generated by Erd"os sets based on coprimality.

Abstract

We study the complexity of $\mathscr{B}$-free subshifts which are proximal and of zero entropy. Such subshifts are generated by Behrend sets. The complexity is shown to achieve any subexponential growth and is estimated for some classical subshifts (prime and semiprime subshifts). We also show that $\mathscr{B}$-admissible subshifts are transitive only for coprime sets $\mathscr{B}$ which allows one to characterize dynamically the subshifts generated by the Erd\"os sets.