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 $1<\alpha <(\sqrt{21}+4)/5\approx 1.717$, 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 $1<\alpha \le 4/3$.

29 pages; modified the first paragraph of the proof of Theorem 1.3 and revised Section 1 slightly