Abstract

We study the incrementation of the lower asymptotic density $\underset{¯}{d}(A+H)$ compared with $\underset{¯}{d}(A)$, where $H\subset \mathbb{N}$ is a specific set and $A\subset \mathbb{N}$ is arbitrary. Ruzsa proved optimal inequalities of $\underset{¯}{d}(A+H)$ for $H$ being a set of prime powers or integer powers. We generalize Ruzsa’s result to sets of polynomial values. Moreover, for $h\in \mathbb{Z}[x]$ with a positive leading coefficient and $H^{\prime }=\{h(p):\text{ prime }p\}$, we prove that $\underset{¯}{d}(A+H^{\prime })>\underset{¯}{d}(A)$ if and only if $h(p)-h(2)$ is not identically zero modulo any $m\ge 2$.

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