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

Abstract

For $α>1$, set $β=1/(α-1)$. We show that, for every $1<α<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^α}\rfloor - \lfloor{m^α}\rfloor$ is equal to $βα^{-β}ζ(β)d^{β-1} + o(d^{β-1})$ as $d\to\infty$, where $ζ$ 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 $\lfloor{l^α}\rfloor + \lfloor{m^α}\rfloor = \lfloor{n^α}\rfloor$. Furthermore, we prove that the additive energy of the sequence $(\lfloor{n^α}\rfloor)_{n=1}^N$, i.e., the number of quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $\lfloor{n_1^α}\rfloor+\lfloor{n_2^α}\rfloor=\lfloor{n_3^α}\rfloor+\lfloor{n_4^α}\rfloor$ and $n_1,n_2,n_3,n_4\le N$, is equal to $O_α(N^{4-α})$ when $1<α\le4/3$.