Difference Sets and Polynomials

TL;DR

This paper establishes upper bounds on the size of subsets of natural numbers avoiding certain polynomial difference patterns, using Fourier analysis and circle method techniques, improving previous results for single polynomial cases.

Contribution

It introduces new bounds for difference sets avoiding polynomial forms, extending and refining earlier work with improved techniques and results.

Findings

Sets free of polynomial difference patterns have exponentially small density.

The methods adapt Fourier analysis and circle method strategies.

Results improve or recover previous bounds for single polynomial cases.

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 UPDATE: While the results and proofs in this preprint are correct, the main result (Theorem 1.1) has been superseded prior to publication by a new paper ( https://arxiv.org/abs/1612.01760 ) that provides better results with considerably less technicality, to which the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.