For each prime , let denote a collection of residue classes modulo such that the cardinalities are bounded and about 1 on average. We show that for sufficiently large , the sifted set contains gaps of size depends only on the densitiy 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 .
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. 667–700DOI 10.4171/JEMS/1020