A maximal extension of the best-known bounds for the Furstenberg–Sárközy theorem

DOI10.4064/AA170828-26-8zbMATH Open1441.11025arXiv1612.01760OpenAlexW2571596282MaRDI QIDQ4614202

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Publication date: 30 January 2019

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Abstract: We show that if $hinmathbbZ[x]$ is a polynomial of degree $kgeq2$ such that $h(mathbbN)$ contains a multiple of $q$ for every $qinmathbbN$, known as an $extitintersectivepolynomial$, then any subset of $1,2,dots,N$ with no nonzero differences of the form $h(n)$ for $ninmathbbN$ has density at most a constant depending on $h$ and $c$ times $(logN{)}^{-cloglogloglogN}$, 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 $g,hinmathbbZ[x]$ are intersective, then any set lacking nonzero differences of the form $g(m)+h(n)$ for $m,ninmathbbN$ has density at most $exp(-c(logN{)}^{mu})$, where $c=c(g,h)>0$, $mu=mu(extdeg(g),extdeg(h))>0$, and $mu(2,2)=1/2$. We also include a brief discussion of sums of three or more polynomials in the final section.

Full work available at URL: https://arxiv.org/abs/1612.01760

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