Diophantine inequalities of fractional degree

TL;DR

This paper studies Diophantine inequalities involving fractional powers, providing asymptotic formulas for solutions and exploring representations of large numbers by generalized polynomials with fractional degrees.

Contribution

It introduces new asymptotic formulas for solutions to fractional degree inequalities and advances mean value estimates for exponential sums with fractional powers.

Findings

Asymptotic formula for solutions within a box of side P

Representation results for large positive numbers

Optimal mean value estimates for exponential sums

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.