Sums of transcendental dilates

TL;DR

The paper proves a lower bound on the size of sumsets involving a finite set and a transcendental dilation, showing it grows faster than any polynomial but slower than exponential, with optimality up to a constant.

Contribution

It establishes a sharp exponential lower bound for sumsets with transcendental dilates, advancing understanding of additive combinatorics involving transcendental numbers.

Findings

Lower bound of exponential form for sumsets with transcendental dilates

Bound is optimal up to a constant factor

Extends additive combinatorics to transcendental dilations

Abstract

We show that there is an absolute constant $c>0$ such that $|A+\lambda\cdot A|\geq e^{c\sqrt{\log |A|}}|A|$ for any finite subset $A$ of $\mathbb{R}$ and any transcendental number $\lambda\in\mathbb{R}$. By a construction of Konyagin and Laba, this is best possible up to the constant $c$.