Additive relations in irrational powers
Joseph Harrison
TL;DR
The paper analyzes the additive structure of irrational power sequences S = {n^c} and shows that for c not in {0,1,2}, the sumset S+S attains its maximal natural size |S+S| ~ N(N+1)/2 as N grows. The authors combine a functional-transcendence approach (Ax–Schanuel) with o-minimal point counting (Pila–Wilkie and a Pfaffian-structure variant) to obtain a uniform bound on the irrational-exponent additive energy, and they derive an effectively computable bound with a computable exponent α. They also establish a Diophantine-approximation criterion ensuring that nontrivial relations among irrational powers are rare, yielding infinitely many effectively computable c for which sets like {p^c} are linearly independent over Q. In the rational-case analysis, determinant-method techniques (Salberger) yield explicit bounds and asymptotics for several exponents, highlighting how the irrational and rational cases differ in their Diophantine geometry. Overall, the work resolves the irrational exponent case and provides computable, structurally informed results about additive dissociation for prime powers as well as concrete bounds in the rational setting.
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 $c$.