On the number of representations of integers as differences between Piatetski-Shapiro numbers

Abstract

For $\alpha>1$, set $\beta=1/(\alpha-1)$. We show that, for every $1<\alpha<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^\alpha}\rfloor - \lfloor{m^\alpha}\rfloor$ is equal to $\beta\alpha^{-\beta}\zeta(\beta)d^{\beta-1} + o(d^{\beta-1})$ as $d\to\infty$, where $\zeta$ denotes the Riemann zeta function. We use this result to derive an asymptotic formula for the number of triplets $(l,m,n)$ of positive integers such that $l<x$ and $\lfloor{l^\alpha}\rfloor + \lfloor{m^\alpha}\rfloor = \lfloor{n^\alpha}\rfloor$. Furthermore, we prove that the additive energy of the sequence $(\lfloor{n^\alpha}\rfloor)_{n=1}^N$, i.e., the number of quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $\lfloor{n_1^\alpha}\rfloor+\lfloor{n_2^\alpha}\rfloor=\lfloor{n_3^\alpha}\rfloor+\lfloor{n_4^\alpha}\rfloor$ and $n_1,n_2,n_3,n_4\le N$, is equal to $O_\alpha(N^{4-\alpha})$ when $1<\alpha\le4/3$.