Large sets avoiding infinite arithmetic / geometric progressions

Abstract

We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen arXiv:2205.04786, we construct a closed set $F\subset[0,\infty)$ such that the measure of $F\cap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou arXiv:2208.02637 for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.