Polynomial values modulo primes on average and sharpness of the larger sieve

TL;DR

This paper investigates the size of subsets of integers with restricted residue classes modulo primes, improving bounds under certain conjectures by analyzing polynomial value sets and their average modulo prime behavior.

Contribution

It introduces an improved bound on the size of such subsets assuming an inverse sieve conjecture, based on studying polynomial value sets modulo primes.

Findings

Bound |X| ≪_α N^{α} from Gallagher's larger sieve.

Under conjecture, |X| can be bounded by N^{O(α^{2014})}.

Average size of polynomial value sets modulo primes influences bounds.

Abstract

This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $\alpha\in (0,1/2)$. Suppose that $|X\pmod p|\leq (\alpha+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\ll_{\alpha}N^{\alpha}$ from Gallagher's larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to $|X|\ll_{\alpha}N^{O(\alpha^{2014})}$ for small $\alpha$). The result follows from studying the average size of $|X\pmod p|$ as $p$ varies, when $X=f(\mathbb{Z})\cap [N]$ is the value set of a polynomial $f(x)\in\mathbb{Z}[x]$.