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}) = \lambda_1 x_1^\theta + \cdots + \lambda_s x_s^\theta. \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 (\textbf{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.