LAUSR

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) = λ1x θ 1 + · · ·+ λsx θ s . 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 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.