Geometric progression-free sequences with small gaps

JOURNAL OF NUMBER THEORY

卷:151

Article

He, Xiaoyu

关键词: Geometric progression-free sequence;    Probabilistic method;    Squarefree gaps;    Divisor function;   

DOI  :  10.1016/j.jnt.2014.12.018

来源: Elsevier

【 摘 要 】

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 Beiglbock 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 epsilon > 0 the interval [x, x + C-epsilon exp((C + epsilon) log x/log log x)] contains an element of T, where C = 5/6 log 2 and C-epsilon > 0 is a constant depending on epsilon. 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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free