DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE. Issue 4 (10th September 2021)

• Record Type: Journal Article
• Authors: Poulias, Constantinos
• Is Part Of: Mathematika. Volume 67:Issue 4(2021)
• Page Start: 949
• Page End: 980
• Publication Date: 2021-09-10
• Subjects: 11D75 (primary) -- 11D72 -- 11P55 -- 11L07 (secondary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12112
• Languages: English
• ISSNs: 0025-5793

Abstract

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 λ i not all of the same sign, we write F ( x ) = λ 1 x 1 θ + ⋯ + λ s x s θ . 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.