Multiplicative group generated by quotients of integral parts

Abstract

For fixed positive numbers \(\alpha \ne \beta \), we consider the multiplicative group \({\mathcal {F}}_{\alpha ,\beta }\) generated by the rational numbers of the form \(\frac{\lfloor \alpha n\rfloor }{\lfloor \beta n\rfloor }\), where \(n \in {\mathbb {N}}\) and \(n \ge \max \big (\frac{1}{\alpha },\frac{1}{\beta }\big )\). For \(0<\alpha <1\) and \(\beta =1\), we prove that \({\mathcal {F}}_{\alpha ,1}\) is the maximal possible group, namely, it is the multiplicative group of all positive rational numbers \({\mathbb {Q}}^{+}\). The same equality \({\mathcal {F}}_{\alpha ,\beta }={\mathbb {Q}}^{+}\) holds in the case when \(0< 10 \alpha \le \beta <1\). These results produce infinitely many pairs \((\alpha ,\beta )\), \(\alpha \ne \beta \), for which a conjecture posed by Kátai and Phong can be confirmed. The only previously known such example is \((\alpha ,\beta )=(\sqrt{2},1)\). We also give a new proof in that case, which is much simpler than the original proof of Kátai and Phong. More generally, we conjecture that \({\mathcal {F}}_{\alpha ,\beta }={\mathbb {Q}}^{+}\) for \(\alpha \ne \beta \) if at least one of the numbers \(\alpha ,\beta >0\) is not an integer.