Geometric progressions in syndetic sets

TL;DR

This paper investigates the presence of geometric progressions within syndetic sets, establishing new results about the types of ratios such sets must contain, including progressions with ratios involving prime powers and perfect squares.

Contribution

It proves that syndetic sets contain 2-term geometric progressions with ratios of specific algebraic forms, advancing understanding of multiplicative structures in additively large sets.

Findings

Syndetic sets contain 2-term geometric progressions with ratios of the form n^k r_1 and p^k r_2.

2-syndetic sets contain infinitely many 2-term geometric progressions with ratios that are perfect squares.

The results extend known cases and answer open questions about 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.