Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics

Department Colloquium

Dimitris Koukoulopoulos (Universite de Montreal)

Friday, January 10th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Dimitris Koukoulopoulos (Université de Montréal)

Title: On the Duffin-Schaeffer conjecture.

Abstract: Given any real number $\alpha$, Dirichlet proved that there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le 1/q^2$. Can we get closer to $\alpha$ than that? For certain "quadratic irrationals" such as $\alpha=\sqrt{2}$ the answer is no. However, Khinchin proved that if we exclude such thin sets of numbers, then we can do much better. More precisely, let $(\Delta_q)_{q=1}^\infty$ be a sequence of error terms such that $q^2\Delta_q$ decreases. Khinchin showed that if the series $\sum_{q=1}^\infty q\Delta_q$ diverges, then almost all $\alpha$ (in the Lebesgue sense) admit infinitely many reduced rational approximations $a/q$ such that $|\alpha-a/q|\le \Delta_q$. Conversely, if the series $\sum_{q=1}^\infty q\Delta_q$ converges, then almost no real number is well-approximable with the above constraints. In 1941, Duffin and Schaeffer set out to understand what is the most general Khinchin-type theorem that is true, i.e., what happens if we remove the assumption that $q^2\Delta_q$ decreases. In particular, they were interested in choosing sequences $(\Delta_q)_{q=1}^\infty$ supported on sparse sets of integers. They came up with a general and simple criterion for the solubility of the inequality $|\alpha-a/q|\le\Delta_q$. In this talk, I will explain the conjecture of Duffin-Schaeffer as well as the key ideas in recent joint work with James Maynard that settles it.

Dimitris Koukoulopoulos is an Associate Professor of Mathematics at the Université de Montréal. He received his PhD from the University of Illinois in 2010. He works in analytic number theory, especially probabilistic and multiplicative aspects of the subject. Among his accolades, he was the cowinner of the 2013 Paul R. Halmos - Lester R. Ford Award. He is the author of the recent book The Distribution of Prime Numbers, published by the AMS.