Distribution of integers with prescribed arithmetic structure and applications

Abstract

This thesis contains results about the distribution of integers with prescribed arithmetic structure and an application. These include a counting problem in Diophantine approximation, an asymptotic formula for the number of solutions to congruence's with certain arithmetic conditions, lower bounds on the number of smooth square-free integers in arithmetic progression, an estimate on the smallest square-full number in almost all residue classes modulo a prime, a relaxation of Goldbach's conjecture from the point of view of Ramare's local model, and lastly a refinement of the classical Burgess bound.