Geometric progression-free sequences with small gaps

Abstract

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 + C ε exp((C + ε) log x/ log log x)] contains an element of T , where C = 5 6 log 2 and C ε > 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(−d k (n)) in very short intervals, where d k (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.

Primitive and geometric-progression-free sets without large gaps

“…There has been partial progress toward this question for 2-syndetic sets (sets in which the difference between any two consecutive terms is at most two). He [11] shows by a computer search that any subset of the range [1,640] containing at least one of any pair of consecutive numbers must contain three term geometric progressions. Recently Patil [17] shows that any sequence of integers s 1 < s 2 < · · · with s n − s n−1 ≤ 2 must contain infinitely pairs {a, ar 2 } with r an integer.…”

“…He [11] considers the existence of geometric-progression-free sets with gaps provably smaller than the bounds for k-free numbers. He shows the following.…”