Sets of full measure avoiding Cantor sets

TL;DR

This paper constructs specific thin symmetric Cantor sets and a full measure set that avoid containing any affine copy of these sets, providing new insights into the Erdős similarity problem.

Contribution

It introduces probabilistic methods to build sets of full measure avoiding affine copies of certain Cantor sets, advancing understanding of measure-theoretic avoidance.

Findings

Constructed symmetric Cantor sets with almost doubly exponential decreasing intervals.

Created a full measure set avoiding any affine copy of the Cantor sets.

Demonstrated probabilistic techniques for measure avoidance in real analysis.

Abstract

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are not universal in measure, i.e. they satisfy the above conjecture. These are symmetric Cantor sets $C$ which can be quite thin: the length of the $n$-th generation intervals defining the Cantor set is decreasing almost doubly exponentially. Further, we achieve to construct a set, not just of positive measure, but of \textit{full measure} not containing any affine copy of $C$. Our method is probabilistic.