We explore the occurrence of point configurations in non-meager 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 \(\mathbb {R}^d\) for \(d\ge 1\) . A topological analogue attributed to Piccard asserts that both AB and \(AB^{-1}\) contain an interval when A, B are non-meager 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 \(P=\{v^i\}_{i\in \mathbb {N}}\) is a bounded sequence, then there is an interval of scalings t for which \(tP+z\subset A\) for some \(z\in \mathbb {R}^d\) . That is, the set \(\Delta _P(A)=\{t>0: \exists z{\text { such that }}tP+z\subset A\}\) has nonempty interior. More generally, if V is a topological vector space and \(P=\{v^i\}_{i\in \mathbb {N}}\subset V\) is a bounded sequence, we show that if \(A\subset V\) is non-meager and Baire, then \(\Delta _P(A)\) has nonempty interior. The notion of boundedness in this context is described below. Note that the sequence \(P\) 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.
