Point configurations in sets of sufficient topological structure and a topological {E}rdős similarity conjecture

Abstract

We explore the occurrence of point configurations within non-meager (second category) Baire sets. A celebrated result of Steinhaus asserts that $A+B$ and $A-B$ contain an interval whenever $A$ and $B$ are sets of positive Lebesgue measure in $\mathbb{R}^n$ for $n\geq 1$. A topological analogue attributed to Piccard asserts that both $AB$ and $AB^{-1}$ contain an interval when $A,B$ are non-meager (second category) Baire sets in a topological group. We explore generalizations of Piccard's result to more complex point configurations and more abstract spaces. In the Euclidean setting, we show that if $A\subset \mathbb{R}^d$ is a non-meager Baire set and $F=\{x_n\}_{n\in\mathbb{N}}$ is a bounded sequence, then there is an interval of scalings $t$ for which $tF+z\subset A$ for some $z\in \mathbb{R}^d$. That is, the set $$Δ_F(A)=\{t\in\mathbb{R}: \exists z\text{ such that }tF+z\subset A\}$$ has nonempty interior. More generally, if $V$ is a topological vector space and $F=\{x_n\}_{n\in\mathbb{N}} \subset V$ is a bounded sequence, we show that if $A\subset V$ is non-meager and Baire, then $Δ_F(A)$ has nonempty interior. The notion of boundedness in this context is described below. Note that the sequence $F$ can be countably infinite, which distinguishes this result from its measure-theoretic analogue. In the context of the topological version of Erdős' similarity conjecture, we show that bounded countable sets are universal in non-meager Baire sets.