Additive relations in irrational powers

Abstract

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 $c \not \in \{0, 1, 2\}$, the cardinality of the sumset $S + S$ asymptotically attains its natural upper bound $N(N + 1)/2$, as $N \to \infty$. We show that there are infinitely many, effectively computable numbers $c$ such that the set $\{p^c : \textrm{$p$ prime}\}$ is additively dissociated (actually linearly independent over $\mathbb{Q}$), and we provide an effective procedure to compute the digits of such $c$.