Geometric progressions in syndetic sets

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}$$,asyndeticsetcontains$2$-term geometric progressions with common ratios of the form $n^k{r}_1$ and $p^k{r}_2$, where $p\in \mathbb{P}$ (the set of primes), $n$ is a composite number, $r_1\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}n)$, $r_2\equiv 1\pmod{p}$and$r_1,r_2\in $\mathbb{N}$$. 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.