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

We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k\ge 2$ such that $h(\mathbb{N})$ contains a multiple of $q$ for every $q\in $\mathbb{N}$$,knownasan$$\mathit{\text{intersective}}\mathit{\text{polynomial}}$$,thenanysubsetof${1,2,\dots,N}$withnononzerodifferencesoftheform$h(n)$for$n\in$\mathbb{N}$$hasdensityatmostaconstantdependingon$h$and$c$times$(\log N)^{-c\log\log\log\log N}$,forany$c<(\log((k^2+k)/2))^{-1}$. 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 $\mu (2,2)=1\mathrm{/}2$. We also include a brief discussion of sums of three or more polynomials in the final section.