Simultaneous equations and inequalities

Abstract

Let ${\lambda }_i,{\mu }_j$ be non-zero real numbers not all of the same sign and let $a_i,b_k$ be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape \begin{equation*} \begin{cases} \left| λ_1 x_1^θ+ \cdots + λ_\ell x_\ell^θ+ μ_1 y_1^θ+ \cdots + μ_m y_m^θ\right| < τ\\[10pt] a_1 x_1^d + \cdots a_\ell x_\ell^d + b_1 z_1^d + \cdots + b_n z_n^d =0, \end{cases} \end{equation*} where $d\ge 2$ is an integer, $\theta >d+1$ is real and non-integral and $\tau$ is a positive real number. For such systems we obtain an asymptotic formula for the number of positive integer solutions $(\text{x},\text{y},\text{z})=(x_1,\dots ,z_n)$ inside a bounded box. Our approach makes use of a two-dimensional version of the classical Hardy-Littlewood circle method and the Davenport--Heilbronn--Freeman method. The proof involves a combination of essentially optimal mean value estimates for the auxiliary exponential sums, together with estimates stemming from the classical Weyl and Weyl-van der Corput inequalities.

Submitted for publication