Filaseta This award funds the continued work of Professor Michael Filaseta as he refines the new differencing techniques he developed in joint work with Prof. Ognian Trifonov. Prof. Filaseta has several specific problems on which he will apply the method. These problems include questions about the distribution of integer values of fairly general functions modulo one, and questions about the gaps between squarefree numbers and gaps between squarefull numbers. This work is in the general field of analytic number theory. The field of analytic number theory applies the techniques of calculus to the discrete realm of the whole numbers. Calculus was developed to aid the analysis of continuous mathematical objects. The idea of using continuous methods to investigate discrete objects is two centuries old, but with the work of the modern analytic number theorists, the field has had a new rebirth and led to a deeper understanding of the whole numbers. ***
