Mean Value Estimates for a Real-Exponent Analogue of Waring's Problem

Abstract

For non-integer $\theta > 3$ and $\kappa \geq 1$, we show that the smallest $r_0$ such that the mean value estimate \[ \int_{-\kappa}^{\kappa} \Big| \sum_{X < x \leq 2X} e(\alpha x^{\theta}) \Big|^{2r} d\alpha \ll_{\epsilon} \kappa X^{2r - \theta+\epsilon} \] holds for all integers $r \geq r_0$ satisfies $2r_0 \leq \theta^2(1+O(\theta^{-1/2}))$. This is an improvement over the previous bound by Poulias of $2r_0 \leq (\lfloor 2\theta \rfloor + 1)(\lfloor 2\theta \rfloor + 2)$. As a consequence, the bound on the asymptotic order of the minimum number of variables required to prove the expected asymptotic formula for the number $R_{s,\theta}(N)$ of solutions $(x_1,\ldots,x_s) \in \mathbb{N}^s$ to the Diophantine equation \[ \lfloor x_1^{\theta} \rfloor+\cdots+\lfloor x_s^{\theta} \rfloor = N \] is improved by a factor of $4$. We also discuss a certain Diophantine system which arises naturally from our proof, which may have applications to other counting problems and may be of independent interest.