Kneser's theorem for upper Buck density and relative results

2024/10/17 by Hennecart, Francois · 1 citation
#11B05 #11B13 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.13275

Abstract

Kneser's theorem in the integers asserts that denoting by \underlined the lower asymptotic density, if \underlined(X₁+⋯+X_k)<∑_i=1^k\underlined(X_i) then the sumset X₁+⋯+X_k is 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.

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