Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets

Abstract

Varjú and Yu have given a complete description of non-singleton self-similar sets of uniqueness on the real line $\mathbb R$. In this paper, we first prove an arithmetic product theorem for self-similar $U$-sets $E:$ if $F\subset\mathbb R$ is compact, then $F\cdot E$ is an $M_0$-set if and only if $F$ is uncountable. We next show that this conclusion remains valid for bounded integer-based homogeneous Moran sets. As an application, for every $s\in(0,1]$, there exists a bounded integer-based homogeneous Moran set of uniqueness $K$ such that $\dim_H K=s$ and the self-product $K\cdot K$ is an $M_0$-set. In sharp contrast, for every $s\in[0,1/2]$ we can construct an unbounded integer-based homogeneous Moran $U$-set $E$ with $\dim_H E=s$ such that both $E$ and $E\cdot E$ are $U$-sets, and $1/2$ is the optimal threshold for $U$-sets to be closed under the arithmetic self-product. Moreover, we show that if $E\subset[0,1]$ is a compact set whose self-product $E\cdot E$ is contained in an $H$-set, then $\underline\dim_B E\le1/2$.