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

Abstract

For non-integer $θ> 3$ and $κ\geq 1$, we show that the smallest $r_0$ such that the mean value estimate \[ \int_{-κ}^κ \Big| \sum_{X < x \leq 2X} e(αx^θ) \Big|^{2r} dα\ll_ε κX^{2r - θ+ε} \] holds for all integers $r \geq r_0$ satisfies $2r_0 \leq θ^2(1+O(θ^{-1/2}))$. This is an improvement over the previous bound by Poulias of $2r_0 \leq (\lfloor 2θ\rfloor + 1)(\lfloor 2θ\rfloor + 2)$. 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 $R_{s,θ}(N)$ of solutions $(x_1,\ldots,x_s) \in \mathbb{N}^s$ to the Diophantine equation \[ \lfloor x_1^θ \rfloor+\cdots+\lfloor x_s^θ \rfloor = N \] 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.