On the Affine Sharpness of Heartʼs Density Theorem

Abstract

Let $\{I_n\}_{n=1}^\infty$ be a sequence of pairwise disjoint intervals tending to 0 from the right, such that putting $I_n=(a_n,b_n)$ we have $\lim\inf {\dfrac{|I_n|}{b_n}}=0$. We prove that for every Cantor set $C$ there exists some set $X$ of positive measure, such that for a.e. $x\in X$ there exists a $c\in C$ for which $(x+cI_n)\cap X=\emptyset$ for infinitely many $n$.