Approximation by Random Fractions by Prof. Felipe Ramirez, Wesleyan University, Class of 1960s Speaker, Friday, Nov. 1, 1 – 1:45 pm, Stetson Court Classroom 110
Abstract: In Diophantine approximation, we study the quality of approximations to real numbers by rational numbers, where “quality” is typically measured against the size of the denominator of the rational number. Consequently, one often finds that results depend in part on whether or not we require that our rational numbers be expressed in reduced form. I will discuss a setting where instead of restricting attention to fractions that are reduced, we restrict to randomly chosen collections of “allowed” rationals. The results and problems I will present are random analogues of Khintchine’s theorem and the Duffin-Schaeffer conjecture.