Plünnecke’s Theorem for asymptotic densities

Abstract:

Plünnecke proved that if $B\subseteq \mathbb {N}$ is a basis of order $h>1$, i.e., $\sigma (hB)=1$, then $\sigma (A+B)\geqslant \sigma (A)^{1-\frac {1}{h}}$, where $A$ is an arbitrary subset of $\mathbb {N}$ and $\sigma$ represents Shnirel’man density. In this paper we explore whether $\sigma$ can be replaced by other asymptotic densities. We show that Plünnecke’s inequality above is true if $\sigma$ is replaced by lower asymptotic density $\underline {d}$ or by upper Banach density $BD$ but not by upper asymptotic density $\overline {d}$. The result about $\underline {d}$ has some interesting consequences such as the inequality $\underline {d}(A+P)\geqslant \underline {d}(A)^{2/3}$ for any $A\subseteq \mathbb {N}$, where $P$ is the set of all prime numbers, and the inequality $\underline {d}(A+C)\geqslant \underline {d}(A)^{3/4}$ for any $A\subseteq \mathbb {N}$, where $C$ is the set of all cubes of nonnegative integers. The result about $BD$ generalizes Theorem 3 of a 2001 work of the author by reducing the requirement of $B$ being a piecewise basis to the requirement of $B$ being an upper Banach basis.