Diophantine approximation and prime numbers

Abstract

In the first part of this thesis various problems in diophantine approximation are considered, which generalize well known theorems of Dirichlet and Kronecker. A brief survey is presented in the first chapter, including a discussion on the scope of elementary methods. It is demonstrated here that stronger results are possible by elementary means than have previously been obtained. In the subsequent chapters non-elementary methods are used. Results are proved for fractional parts of quadratic forms in several variables which improve upon previous work. New theorems are demonstrated for the distribution modulo one of 'almost all" additive forms in many variables, including the particularly interesting case of a linear form in positive variables. In chapter four new bounds are given for exponential sums over primes, which greatly improve upon the work of I.M. Vinogradov. Some applications to diophantine approximation problems involving primes are given in chapters 4 and 5, the latter chapter also improving upon previous work on the problem of a linear form in three prime variables. In the second section, topics in multiplicative number theory are discussed. It is shown that almost-primes are very well distributed in almost all very short intervals, improving upon previous work by a considerable factor. Sieve methods are then employed to tackle three other problems. New results are in this way obtained for primes in short intervals, for the distribution of the square roots of primes (modulo one), and for the distribution of [alpha] modulo one for irrational [alpha]. This last chapter contains a new method for tackling sums over primes which has other applications.