Full measure universality for Cantor Sets
Pablo Shmerkin, Alexia Yavicoli
TL;DR
The paper advances the Erdős similarity program by showing that Cantor-type sets are not full measure universal under mild logarithmic-dimension conditions, advancing beyond prior results for sets with positive dimension. It develops a general inductive framework (RRP) combined with a Ruzsa-type combinatorial lemma to force sumsets to acquire nonempty interior and to obstruct universality via measure-zero sets. The authors establish a robust probabilistic/analytic pipeline, including discrete Fourier techniques and local Fourier decay arguments, to transfer finite-group sumset control to the continuous setting and to obtain null-obstruction sets B for broad classes of A and families of maps (affine, polynomial, convex, bi-Lipschitz). The results yield explicit formulations: for Cantor A, there exists a dense full-measure X so that for any bi-Lipschitz f, the translates of f(A) do not occupy positive-measure portions of X, and, more generally, for many A one can ensure R^d is covered by A+B up to a null B for wide families of images, greatly strengthening the non-universality landscape.
Abstract
We investigate variants of the Erdős similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known fact that sets of positive Hausdorff dimension are not measure universal. We prove a weaker result for all Cantor sets $A$: there is a dense $G_δ$ set of full measure $X\subset\mathbb{R}^d$, such that for any bi-Lipschitz function $f:\mathbb{R}^d\to \mathbb{R}^d$, the set of translations $t$ such that $f(A)+t\subseteq X$ is of measure zero. Equivalently, there is a null set $B\subset\mathbb{R}^d$ such that $\mathbb{R}^d\setminus (f(A)+B)$ is null for all bi-Lipschitz functions $f$.