On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs

Abstract

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Kr\'al', Serra and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct application, we obtain the following generalization of Szemer\'edi's theorem: for any compact abelian group $G$, any measurable set $A\subset G$ with Haar probability $\mu(A)\geq\alpha>0$ satisfies $$\int_G\int_G\; 1_A\big(x\big)\; 1_A\big(x+r\big) \cdots 1_A\big(x+(k-1)r\big) \; d\mu(x) d\mu(r) \geq c,$$ where the constant $c=c(\alpha,k)>0$ is valid uniformly for all $G$. This result is shown to hold more generally for any translation-invariant system of $r$ linear equations given by an integer matrix with coprime $r\times r$ minors.