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 $\{n\in \mathbb{Z}:n(\mathrm{mod}p)\notin I_{p}for\; allp\le x\}$ contains gaps of size at least $x(\log x{)}^{\delta }$ where $\delta >0$ depends only on the density of primes for which $I_p\ne \varnothing$. 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 $\{n\le X:f(n)composite\}$ 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 $f$.

Corrects a number of errors in the published version pointed out to us by Mikhail Gabdullin. The specific corrections are listed in an Appendix. One of these causes a slight degradation of the numerical values for the exponents of log log x in Theorem 1 and Corollaries

References in corpus (1)

• A coprimality condition on consecutive values of polynomials

Cited by in corpus (1)

• A large integer is a sum of two prime avoiding numbers