Difference Sets and Polynomials

Abstract

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 in the classes of so-called intersective and $\mathcal{P}$-intersective polynomials, respectively. 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 $e^{-(\log N)^{\mu}}$ for some $\mu=\mu(j,k)>0$. Our results, obtained by adapting two Fourier analytic, circle method-driven strategies, either recover or improve upon all previous results for a single polynomial. UPDATE: While the results and proofs in this preprint are correct, the main result (Theorem 1.1) has been superseded prior to publication by a new paper ( https://arxiv.org/abs/1612.01760 ) that provides better results with considerably less technicality, to which the interested reader should refer.