An analytic approach to some diophantine inequalities

Heilbronn proved that for any epsilon > 0 there exists a number C (epsilon) such that for any real numbers theta and N(≥ 1) min n2 theta-1/2+epsilon In the first part of this thesis we prove various extensionsof this result. We find values of g(r, k, s) so that the inequality [equation] is soluble, where X is an integral s-dimensional vector and the f's are either polynomials (without constant term) or forms, both of degree ≤1 k. The method used depends upon estimates for certain exponential sums. Using Weyl's estimates, we look, in Chapter 2, at monomials of different degree, and, in Chapter 3, at additive forms of degree k in s variables and quadratic polynomials. Using Hua's improvement of Vinogradov's estimates, we improve, for large values of k, the results of Chapter 2 and the results of Chapter 3 on additive forms. Also using Hua's estimates, we look, in Chapter 6 at polynomials of any degree (≤ k). In the course of this work we improve some results of Liu and Cook. Birch, Davenport and Ridout proved that if Q is an indefinite quadratic form in n( ≥ 21) variables, the inequality |Q (X)| infinity) f(n, k) = 1/2, (ii) for k ≥ 7, 10k 3n lim f(n, k) = ½, and (iii) for 10k ≥ 3n, lim(N->infinity) f(n, k) = 1/2, k->infinity