Primitive and geometric-progression-free sets without large gaps

TL;DR

This paper demonstrates the existence of primitive and geometric-progression-free sets with smaller gaps than prime numbers, using probabilistic methods to improve known bounds.

Contribution

It introduces new probabilistic constructions that establish smaller gap bounds in primitive and geometric-progression-free sets compared to previous results.

Findings

Primitive sets with smaller gaps than primes

Improved bounds for geometric-progression-free sets

Probabilistic methods effectively establish new gap bounds

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.