Expander estimates for cubes

Abstract: Suppose that $\mathcal A$ is a subset of the natural numbers. The supremum $\alpha$ of all $t$ with $$ \limsup N^{-t} \#\{a\in{\mathcal A}: a\le N\} >0 $$ is the {\em exponential density} of $\mathcal A$.

We examine what happens if one adds a power to $\mathcal A$. Fix $k\ge 2$, and let $\beta_k$ be the exponential density of $$ \{ x^k+a : x\in {\mathbb N}, \, a\in{\mathcal A}\}.$$ It is easy to see that $\beta_2= \min (1, \frac12 +\alpha).$ One might guess that $$ \beta_k = \min (1, \frac{1}{k}+\alpha) \eqno (*) $$ holds for all $k$, but we are far from a proof. All current world records for this problem are due to Davenport, and are 80 years old. In this interim report on ongoing work with Simon Myerson, we describe a method for $k=3$ that improves Davenport's results when $\alpha>3/5$, and that confirms (*) in an interval $(\alpha_0, 1]$. A concrete value for $\alpha_0$ will be released during the talk, and if time permits, we also discuss the perspectives to generalize the approach to larger values of $k$.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)