Kneser's theorem for upper Buck density and relative results

TL;DR

This paper extends Kneser's theorem to the context of upper Buck density, comparing it with upper Banach density results, and constructs sequences with counterintuitive properties related to sumsets and densities.

Contribution

It establishes a Kneser-type theorem for upper Buck density and compares it with Jin's result on upper Banach density, including novel sequence constructions.

Findings

Kneser's theorem holds for upper Buck density with periodic sumsets.

Comparison between Buck density and Banach density results.

Construction of sequences with counterintuitive Buck density properties.

Abstract

Kneser's theorem in the integers asserts that denoting by $ \underline{\mathrm{d}}$ the lower asymptotic density, if $\underline{\mathrm{d}}(X_1+\cdots+X_k)<\sum_{i=1}^k\underline{\mathrm{d}}(X_i)$ then the sumset $X_1+\cdots+X_k$ is \emph{periodic} for some positive integer $q$. In this article we establish a similar statement for upper Buck density and compare it with the corresponding result due to Jin involving upper Banach density. We also provide the construction of sequences verifying counterintuitive properties with respect to Buck density of a sequence $A$ and its sumset $A+A$.