Additive relations in irrational powers

Abstract

We investigate the interaction between raising to an irrational power and addition of real numbers. Thus, for a finite set $A$ of non-negative real numbers, let $A^{[c]} = \{a^c : a \in A\}$. When $k$ is a positive integer, $c$ is a real irrational number, and $A$ is a subset of an $N$-term arithmetic progression in $\mathbb{R}_{\geq 0}$ having cardinality at least a power of $\log{N}$, we prove that the $k$-fold sumset $|kA^{[c]}| \sim_k |A|^k/k!$ as $|A| \to \infty$. This result is uniform in $c$. When $A = \{1, \dots, N\}$ and $k = 2$, this result can be combined with existing works to show that $|A^{[c]} + A^{[c]}| \sim N^2/2$ as $N \to \infty$ whenever $c \in \mathbb{R} \setminus \{0, 1, 2\}$. The sumset lower bound follows from a bound on the number of equal sums of $r$ and $s \geq r$ elements of $A^{[c]}$ (by taking $r = s = k$). When $r = s = 2$ or $s > r$, our bound is optimal up to a power of $\log N$. This bound is proved using a functional transcendence theorem for certain endomorphisms of $\mathbb{R}_{>0}^n$, and innovations in the Pila--Wilkie counting theorem in $\mathbb{R}_{\exp}$ due to Binyamini, Novikov and Zak. In a different direction, we provide a Diophantine approximation criterion on $c$ that, when satisfied, ensures that a linear form in the $c$-th powers of multiplicatively independent integers does not vanish. The proof involves linear forms in logarithms. This provides a new proof of a fact, due to Bays--Kirby--Wilkie and Jones--Servi, that when $A$ is a multiplicatively independent set of positive integers, there are infinitely many effectively computable real numbers $c$ such that $A^{[c]}$ is linearly independent over $\mathbb{Q}$.