Primitive and geometric-progression-free sets without large gaps

Abstract

We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.2010 Mathematics Subject Classification. 11N25 (primary), and 11B05 (secondary).