Diophantine and Metric Number Theory

Diophantine number theory, named after the ancient Greek mathematician Diophantus of Alexandria, revolves around problems concerning the natural numbers (i.e., the numbers 1, 2, 3, ...). Such problems have been puzzling humans for thousands of years. Often very simple questions turn out to be surprisingly difficult to answer. For example, the following is an open question: Are 25 and 27 the only perfect powers with distance exactly 2? Here by "perfect powers" we mean squares, cubes, and so on (i.e., the numbers 4, 8, 9, 16, 25, ). Metric number theory, on the other hand, is concerned with questions of the following type: "How large is the set of real numbers with some given property?" It turns out that often one can make statements about properties of "almost all numbers", that are very hard to establish for a single chosen number. For example, it is still not exactly known how "efficiently" the number pi can be approximated by rationals. However, for "almost all numbers" this can be determined in a very precise way, in particular since the recent proof of the Duffin-Schaeffer conjecture. In this project, we want to combine questions and methods from both fields of number theory, and thus further our understanding of properties of numbers. Our specific goals include the following: First, we want to better understand the distribution of gaps between certain powers. Secondly, to make progress on so-called shrinking target problems. And thirdly, to investigate in more detail how well "random behaviour" can be imitated through an interplay of rapidly growing sequences of integers and single real numbers. Throughout the project, the approximability of real numbers by rationals will play a crucial role.