Solutions to several problems on infinite sumset configurations

Abstract

We resolve a number of the problems collected by Kra, Moreira, Richter and Robertson in their survey "Problems on infinite sumset configurations in the integers and beyond". Our main result is that in every countably infinite abelian group, every set of positive upper Banach density contains a sumset $B_1+\dots+B_k$ of $k$ infinite sets, for every $k$; the proof uses the Kronecker factor, but no higher-order Host-Kra structure theory, nilsystems of step greater than one, or nilspaces. We also answer, positively or negatively, questions on monochromatic sets $B+mB$, on sumsets in sets of logarithmic density, on polynomial sumset configurations, on sums of the form $B_{i_1}+\dots+B_{i_m}$ with $B_i$ drawn from a density family, on random sets, on ordered sumsets in shifted primes, on sums and products in $\mathbb{N}$ and $\mathbb{Q}$, and on sumsets in abelian groups. Several of these answers concern the questions as literally stated, and two follow directly from known examples; in each case we say so and discuss the natural corrected question. Finally, answering a question of Kousek and Radić, we show that the supremum of the lower Banach densities of sets $A\subset\mathbb{N}$ such that $B+B+t\not\subset A$ for every infinite $B\subset\mathbb{N}$ and every $t\ge0$ (respectively $B+B\not\subset A$ for every infinite $B$) is $1/2$ (respectively $3/4$). The main theorem and most of the other results are formalized in Lean 4, with some published theorems entering as explicit hypotheses.