A Maximal Extension of the Best-Known Bounds for the Furstenberg-S\'ark\"ozy Theorem

Abstract

We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k \geq 2$ such that $h(\mathbb{N})$ contains a multiple of $q$ for every $q\in \mathbb{N}$, known as an $\textit{intersective polynomial}$, then any subset of $\{1,2,\dots,N\}$ with no nonzero differences of the form $h(n)$ for $n\in\mathbb{N}$ has density at most a constant depending on $h$ and $c$ times $(\log N)^{-c\log\log\log\log N}$, for any $c<(\log((k^2+k)/2))^{-1}$. Bounds of this type were previously known only for monomials and intersective quadratics, and this is currently the best-known bound for the original Furstenberg-S\'ark\"ozy Theorem, i.e. $h(n)=n^2$. The intersective condition is necessary to force any density decay for polynomial difference-free sets, and in that sense our result is the maximal extension of this particular quantitative estimate. Further, we show that if $g,h\in \mathbb{Z}[x]$ are intersective, then any set lacking nonzero differences of the form $g(m)+h(n)$ for $m,n\in \mathbb{N}$ has density at most $\exp(-c(\log N)^{\mu})$, where $c=c(g,h)>0$, $\mu=\mu(\text{deg}(g),\text{deg}(h))>0$, and $\mu(2,2)=1/2$. We also include a brief discussion of sums of three or more polynomials in the final section.