BILLIARDS AND FAREY FRACTIONS
Sequences of rational numbers with denominators bounded by a given quantity, known as Farey sequences, are very useful tools in many mathematical problems. For example, they play an important role in Diophantine approximation, where one studies the best possible approximations to real numbers by rational numbers; in additive problems such as Goldbach Conjecture; and in multiplicative number theory, via their connection to the Riemann Hypothesis. Recently, Professor Zaharescu has been studying the local spacing distribution of Farey sequences. The results he obtained on the distribution of gaps between consecutive Farey fractions helped to answer some open problems in number theory and mathematical physics related to billiards and the periodic Lorentz gas.
During his Center appointment, he will pursue several projects concerned with the structure of Farey sequences and their connection to billiards and lattice points inside expanding regions. Another goal is to generalize the methods that he and his collaborators have devised over the past few years to make further advances in this area.