Long gaps in sieved sets
Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, Terence, Tao

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 , let denote a collection of residue classes modulo such that the cardinalities are bounded and about on average. We show that for sufficiently large , the sifted set contains gaps of size at least where depends only on the density of primes for which . This improves on the "trivial" bound of . As a consequence, for any non-constant polynomial with positive leading coefficient, the set contains an interval of consecutive integers of length for sufficiently large , where depends only on the degree of .
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