  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
$1<\alpha\le4/3$.
