Difference Sets and Sums of Polynomials

Neil Lyall, Alex Rice

Published 2015-04-20Version 1

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in \mathbb{Z}[x]$ lie in the largest possible classes of polynomials. For example, we show that a subset of $\{1,2,\dots,N\}$ free of nonzero differences of the form $n^j+m^k$ for fixed $j,k\in \mathbb{N}$ has density at most $e^{-c(\log N)^{1/4}}$ for some $c=c(j,k)>0$. Our results, obtained by adapting two Fourier analytic, circle method-driven strategies, either recover or improve upon all previous results for a single polynomial.