In this paper, we confirm several conjectures of Bényi and Ćurgus related to the description of the set \(\textrm{Range}(\alpha )\) attained by the differences of two floor functions \(\lfloor \alpha ^2 n \rfloor \) and \(\lfloor \alpha \lfloor \alpha n\rfloor \rfloor \) , where \(\alpha \) is a fixed positive number and n runs through all positive integers. In particular, we show that for each irrational number \(\alpha >0\) the set \(\textrm{Range}(\alpha )\) is the largest possible and takes all integral values between 0 and \(\lfloor \alpha \rfloor +1\) except possibly for some quadratic algebraic numbers \(\alpha >1\) when \(\textrm{Range}(\alpha )\) can be smaller and take only integral values between 1 and \(\lfloor \alpha \rfloor \) . Moreover, for every quadratic algebraic number \(\alpha >0\) , we explicitly determine the set \(\textrm{Range}(\alpha )\) as well. In both cases, \(t=0\) and \(t=\lfloor \alpha \rfloor +1\) , we give explicit conditions describing whether t is or is not in \(\textrm{Range}(\alpha )\) . Those conditions are given in terms of the coefficients u, v of the minimal polynomial \(x^2-ux-v \in {\mathbb {Q}}[x]\) of \(\alpha \) over \({\mathbb {Q}}\) . All these results allow us to determine the set \(\textrm{Range}(\alpha )\) explicitly for every irrational number \(\alpha >0\) . We also describe all the cases when \(\textrm{Range}(\alpha )\) is a singleton set.
