Kneser's theorem for upper Buck density and relative results

Kneser’s theorem in the integers asserts that denoting by $\underset{‾}{\mathrm{d}}$ the lower asymptotic density, if $\underset{‾}{\mathrm{d}}(X_1+\cdots +X_k)<{\sum }_{i=1}^k\underset{‾}{\mathrm{d}}(X_i)$ 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 $A+A$.