Geometric progressions in syndetic sets

Abstract

In order to investigate multiplicative structures in additively large sets, Beiglb\"{o}ck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains arbitrarily long geometric progressions. A result of Erd\H{o}s implies that syndetic sets contain a $2$-term geometric progression with integer common ratio, but we still do not know if they contain such a progression with common ratio being a perfect square. In this article, we prove that for each $k\in \mathbb{N}$, a syndetic set contains $2$-term geometric progressions with common ratios of the form $n^kr_1$ and $p^kr_2$, where $p\in\mathbb{P}$ (the set of primes), $n$ is a composite number, $r_1\equiv 1 \pmod{n}$, $r_2\equiv 1\pmod{p}$ and $r_1,r_2\in \mathbb{N}$. We also show that 2-syndetic sets (sets with bounded gap two) contain infinitely many $2$-term geometric progressions with their respective common ratios being perfect squares.