Additive relations in irrational powers

TL;DR

This paper studies the additive properties of sets formed by irrational powers, establishing their sumset sizes and additive independence, thus advancing understanding in additive combinatorics for irrational exponents.

Contribution

It resolves the behavior of additive energy for irrational powers and identifies infinitely many additively dissociated irrational powers with an effective computation method.

Findings

Sumset size asymptotically reaches the upper bound for most irrational powers.

Existence of infinitely many additively dissociated irrational powers.

Effective procedure to compute digits of such 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 $c$.