Additive relations in irrational powers

We investigate the additive theory of the set $S=\{1^c,2^c,\dots ,N^c\}$ when $c$ is a real number. In the language of additive combinatorics, we determine the asymptotic behaviour of the additive energy of $S$. When $c$ is rational, this is either known, or follows from existing results, and our contribution is a resolution of the irrational case. We deduce that for all $c\in ̸\{0,1,2\}$, the cardinality of the sumset $S+S$ asymptotically attains its natural upper bound $N(N+1)\mathrm{/}2$, as $N\to \mathrm{\infty }$. We show that there are infinitely many, effectively computable numbers $c$ such that the set \{p^c : \textrm{pprime}\} is additively dissociated (actually linearly independent over $\mathbb{Q}$), and we provide an effective procedure to compute the digits of such $c$.