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 ${\mathbb{R}}_{\ge 0}$ having cardinality at least a power of $\log N$, we prove that the $k$-fold sumset $|k{A}^{[c]}|{\sim }_k|A{|}^k/k!$ 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 $s\ge r$ 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 $\mathbb{Q}$.

20 pages, no figures, comments welcome!