Geometric progression-free sequences with small gaps

Abstract

Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression-free sequence of positive integers. In this paper we prove the existence of geometric progression-free sequences with small gaps, partially answering a question posed originally by Beiglböck et al. Using probabilistic methods we prove the existence of a sequence T not containing any 6-term geometric progressions such that for any x≥1 and ε>0 the interval [x,x+Cεexp⁡((C+ε)log⁡x/log⁡log⁡x)] contains an element of T, where C=56log⁡2 and Cε>0 is a constant depending on ε. As an intermediate result we prove a bound on sums of functions of the form f(n)=exp⁡(−dk(n)) in very short intervals, where dk(n) is the number of positive k-th powers dividing n, using methods similar to those that Filaseta and Trifonov used to prove bounds on the gaps between k-th power free integers.