Finitely additive measures on $\mathbb{Z}$ and additive combinatorics

Abstract

We study (bounded) finitely additive measures on the group of integers $\mathbb{Z}$, as elements of the Banach algebra $\mathrm{ba}(\mathbb{Z})$, viewed as a natural generalization of ultrafilters. The algebraic structure of $\mathrm{ba}(\mathbb{Z})$ extends the semigroup structure of the Čech--Stone compactification, allowing methods from ultrafilter theory to be applied in a broader measure-theoretic setting. We investigate idempotent finitely additive measures and establish additive properties of subsets of $\mathbb{Z}$ having positive measure. We then proceed to study almost translation-invariant and translation-invariant finitely additive measures, showing that these stronger notions yield correspondingly stronger additive conclusions. In particular, we prove that every subset of $\mathbb{Z}$ whose measure exceeds a certain explicit threshold necessarily is an ${\mathsf{IP}}_n$-set; with stronger properties and lower thresholds depending on the properties of the relevant measures. Several examples illustrating the sharpness and limitations of the results are also presented, together with a discussion of open problems and directions for future research.

18 pages

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