Diophantine inequalities of fractional degree

2021/07/30 by Poulias, Constantinos
#11D72 #11D75 #11L07 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2107.14536

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 \mathcal F (x) = λ₁ x₁^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 | \mathcal 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 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.

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