An Approximate Version of Vu's Theorem on Economical Subbases For Non-Integer Exponents

Abstract

We prove an approximate analog of Vu's Theorem on economical bases of $k$th powers for non-integer powers. Fix a non-integer $\theta >2$ and real $\tau >0$. Let $s\ge {\theta }^2+9{\theta }^{3/2}+2$ if $\theta >3$ and $s\ge (\lfloor 2\theta \rfloor +1)(\lfloor 2\theta \rfloor +2)+1$ if $2<\theta <3$. We show the existence of a set $\mathfrak{X}\subseteq \mathbb{N}$ such that the number $R_{\mathfrak{X},s,\theta ,\tau }(\Lambda )$ of integer solutions $(x_1,\dots ,x_s)\in {\mathfrak{X}}^s$ to the equation \begin{align*} |x_1^θ +\cdots+ x_s^θ - Λ| < τ\end{align*} satisfies $R_{\mathfrak{X},s,\theta ,\tau }(\Lambda )\asymp \tau \log (\Lambda )$ for all sufficiently large real $\Lambda >0$.

21 pages