On the Erd\"os similarity problem

TL;DR

This paper advances the understanding of Erd"os's similarity problem by providing new partial results on the existence of positive measure sets avoiding geometric similarity to given infinite sets.

Contribution

It introduces novel partial results addressing Erd"os's longstanding problem about measure and geometric similarity in real sets.

Findings

Established new partial results related to Erd"os's similarity problem

Proved existence of measure-positive sets avoiding certain geometric similarities

Contributed to the understanding of geometric structure in measure theory

Abstract

New partial results are obtained related to the following old problem of Erd\"os: for any infinite set $X$ of real numbers to show that there is always a measurable (or, equivalently, closed) subset of reals of positive Lebesgue measure which contains no subset geometrically similar to $X$.