Additive properties for sets of polynomial values

摘要

We study the incrementation of the lower asymptotic density $underline{dmskip-2mu}mskip2mu(A+H)$ compared with $underline{dmskip-2mu}mskip2mu(A)$, where $Hsubset mathbb{N}$ is a specific set and $Asubsetmathbb{N}$ is arbitrary. Ruzsa proved optimal inequalities of $underline{dmskip-2mu}mskip2mu(A+H)$ for $H$ being a set of prime powers or integer powers. We generalize Ruzsa's result to sets of polynomial values. Moreover, for $hin mathbb{Z}x$ with a positive leading coefficient and $H'={h(p): text{prime }p}$, we prove that $underline{dmskip-2mu}mskip2mu(A+H')underline{dmskip-2mu}mskip2mu(A)$ if and only if $h(p)-h(2)$ is not identically zero modulo any $mgeq 2$.