Kenyon Theorem Revisited

TL;DR

This paper revisits the Kenyon Theorem, providing new conditions for when certain sum sets contain intervals, especially focusing on cases where the set size is prime, expanding understanding of these fractal-like sets.

Contribution

It offers new equivalent conditions for the sum sets to contain intervals under the assumption q|Σ|=1, and provides a complete characterization for prime set sizes.

Findings

New conditions for interval containment in sum sets

Full characterization when set size is prime

Enhanced understanding of sum set structures

Abstract

We study sets $E(\Sigma,q)=\left\{\sum_{i=1}^\infty \sigma_iq^i\colon(\sigma_i)\in\Sigma^{\mathbb N}\right\}$ for a finite set $\Sigma\subset \mathbb R$ and $q\in(0,1)$. Under the assumption $q|\Sigma|=1$ we prove several new equivalent conditions for $E(\Sigma,q)$ to contain an interval. We give a full characterization, if additionally $|\Sigma|$ is prime.