DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE. Issue 4 (10th September 2021) - Poulias, Constantinos
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.