A Maximal Extension of the Best-Known Bounds for the Furstenberg-Sárközy Theorem

Abstract

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 $q\in \mathbb{N}$, known as an $\text{intersective polynomial}$, 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-Sárközy 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.

30 pages, minor revision of published version, small changes to Lemma 4.5, the value of $\mathcal{Q}$ in the proof of Theorem 1.2, and the value of $\mu$ in the statement of Theorem 5.7. arXiv admin note: text overlap with arXiv:1504.04904

Cited by in corpus (1)

• Multivariate Polynomial Values in Difference Sets