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# Journal of the European Mathematical Society

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**Volume 23, Issue 2, 2021, pp. 667–700**

**DOI: 10.4171/JEMS/1020**

Published online: 2020-11-15

Long gaps in sieved sets

Kevin Ford^{[1]}, Sergey Konyagin

^{[2]}, James Maynard

^{[3]}, Carl B. Pomerance

^{[4]}and Terence Tao

^{[5]}(1) University of Illinois at Urbana-Champaign, USA

(2) Steklov Mathematical Institute, Moscow, Russia

(3) University of Oxford, UK

(4) Dartmouth College, Hanover, USA

(5) University of California, Los Angeles, USA

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 $f$.

*Keywords: *Gaps, prime values of polynomials, sieves

Ford Kevin, Konyagin Sergey, Maynard James, Pomerance Carl, Tao Terence: Long gaps in sieved sets. *J. Eur. Math. Soc.* 23 (2021), 667-700. doi: 10.4171/JEMS/1020