Long gaps in sieved sets

TL;DR

The paper demonstrates that sieved sets defined by residue class conditions modulo primes contain arbitrarily large gaps, leading to long intervals of composite values for polynomials, improving known bounds on such gaps.

Contribution

It establishes new lower bounds on the size of gaps in sieved sets and polynomial value sets, advancing understanding of their distribution and structure.

Findings

Existence of gaps of size at least x (log x)^δ in sieved sets

Long intervals of consecutive composite numbers for polynomial values

Improvement over trivial bounds for 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 $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^{\delta} $ where $\delta>0$ depends only on the density of primes for which $I_p\ne \emptyset$. 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 $\{ n \leq X: f(n) \hbox{ composite}\}$ 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 $f$.