# Long gaps in sieved sets

### Kevin Ford

University of Illinois at Urbana-Champaign, USA### Sergey Konyagin

Steklov Mathematical Institute, Moscow, Russia### James Maynard

University of Oxford, UK### Carl B. Pomerance

Dartmouth College, Hanover, USA### Terence Tao

University of California, Los Angeles, USA

## 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 $\{n \in \mathbb{Z}: n (\mathrm {mod}\: p) \not \in I_p \: \mathrm {for \: all} p \leq x\}$ contains gaps of size $x (\log x)^{\delta}$ depends only on the densitiy of primes for which $I_p \neq \emptyset$. 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 $\{n \leq X: f(n) \; \mathrm {composite}\}$ 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 $f$.

## Cite this article

Kevin Ford, Sergey Konyagin, James Maynard, Carl B. Pomerance, Terence Tao, Long gaps in sieved sets. J. Eur. Math. Soc. 23 (2021), no. 2, pp. 667–700

DOI 10.4171/JEMS/1020