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 $x\ge 1$ and $\epsilon >0$ the interval $[x,x+C_{\epsilon }\exp ((C+\epsilon )\log x/\log \log x)]$ contains an element of $T$, where $C=\frac{5}{6}\log 2$ 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.

References in corpus (3)

• Large gaps between consecutive prime numbers
• Long gaps between primes
• Irrational numbers associated to sequences without geometric progressions