Finding product sets in some classes of amenable groups

Abstract

In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$ , for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$ . In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].

Keywords

• 05D10
• 37A15
• 11B13
• 11B30