Long gaps in sieved sets

Abstract

For each prime $p$, let $I_p\subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about 1 on average. We show that for sufficiently large $x$, the sifted set $\{n\in \mathbb{Z}:n(\mathrm{mod}p)\notin I_p\mathrm{for}\mathrm{all}p\le x\}$ contains gaps of size $x(\log x{)}^{\delta }$ depends only on the densitiy of primes for which $I_p\ne \varnothing$. This improves on the "trivial'' bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{n\le X:f(n)\mathrm{composite}\}$ contains an interval of consecutive integers of length $\ge (\log X)(\log \log X{)}^{\delta }$ for sufficiently large $X$, where $\delta >0$ depends only on the degree of $f$.

A correction to this paper is available.