Diophantine inequalities of fractional degree

Abstract

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $\theta ,$ where $\theta >2$ is real and non-integral. For fixed non-zero real numbers ${\lambda }_i$ not all of the same sign we write \begin{equation*} \mathcal F (\textbf{x}) = λ_1 x_1^θ+ \cdots + λ_s x_s^θ. \end{equation*} For a fixed positive real number $\tau$ we give an asymptotic formula for the number of positive integer solutions of the inequality $|\mathcal{F}(\text{x})|<\tau$ inside a box of side length $P.$ Moreover, we investigate the problem of representing a large positive real number by a positive definite generalized polynomial of the above shape. A key result in our approach is an essentially optimal mean value estimate for exponential sums involving fractional powers of integers.

Accepted for publication in Mathematika