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DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE

DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree θ, where θ>2 is real and non‐integral. For fixed non‐zero real numbers λi not all of the same sign, we write F(x)=λ1x1θ+⋯+λsxsθ.For a fixed positive real number τ, we give an asymptotic formula for the number of positive integer solutions of the inequality |F(x)|<τ inside a box of side length P. Moreover, we investigate the problem of representing a large positive real number by a positive definite generalised 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematika Wiley

DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE

Poulias, Constantinos
Mathematika , Volume 67 (4) – Oct 1, 2021
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References (36)

Publisher
Wiley
Copyright
© 2021 London Mathematical Society
ISSN
0025-5793
eISSN
2041-7942
DOI
10.1112/mtk.12112
Publisher site
See Article on Publisher Site

Abstract

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree θ, where θ>2 is real and non‐integral. For fixed non‐zero real numbers λi not all of the same sign, we write F(x)=λ1x1θ+⋯+λsxsθ.For a fixed positive real number τ, we give an asymptotic formula for the number of positive integer solutions of the inequality |F(x)|<τ inside a box of side length P. Moreover, we investigate the problem of representing a large positive real number by a positive definite generalised 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.

Journal

Mathematika – Wiley

Published: Oct 1, 2021

Keywords: 11D75 (primary); 11D72; 11P55; 11L07 (secondary)

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