Exponential sums and Diophantine problems.

Abstract

This work is concerned with the theory of exponential sums and their application to various Diophantine problems. Particular attention is given to exponential sums over smooth numbers, i.e. numbers having no large prime factors. As an application of the theory of exponential sums in a single variable, we consider pairs of Diophantine inequalities of different degrees. Specifically, we show that two additive forms, one cubic and one quadratic, with real coefficients in at least 13 variables and satisfying suitable conditions, take arbitrarily small values simultaneously at integer points. In fact, we obtain a quantitative version of this result, which indicates how rapidly the forms can be made to approach zero as the size of the variables increases. Moreover, we obtain a lower bound for the density of integer points at which these small values occur. We then proceed to study double exponential sums over smooth numbers by developing a version of the Vaughan-Wooley iterative method. We obtain estimates for mean values of these exponential sums, and these estimates are then used within the fabric of the Hardy-Littlewood method to obtain a lower bound for the density of rational lines on the hypersurface defined by an additive equation. We show that one obtains the expected density provided that the number of variables is sufficiently large in terms of the degree and that certain natural local solubility hypotheses are satisfied. We also consider applications to a two-dimensional generalization of Waring's problem and to fractional parts of polynomials in two variables. Finally, we refine the above analysis in the case of a cubic hypersurface to show that the expected density of lines is obtained whenever the defining equation has at least 58 variables. In the process, we obtain a result on the paucity of non-trivial solutions to an associated system of Diophantine equations.