Number Theory

Title: On the Duffin-Schaeffer conjecture
Speaker: Dimitris Koukoulopoulos
Speaker Info: Université de Montréal
Brief Description: (note special time at 2:00pm and place Lunt 102)
Special Note:

Let $S$ be a sequence of integers. We wish to understand how well we can approximate a ``typical'' real number using reduced fractions whose denominator lies in $S$. To this end, we associate to each $q\in S$ an acceptable error $\delta_q>0$. When is it true that almost all real numbers (in the Lebesgue sense) admit an infinite number of reduced rational approximations $a/q$, $q\in S$, within distance $\delta_q$? In 1941, Duffin and Schaeffer proposed a simple criterion to decide whether this is case: they conjectured that the answer to the above question is affirmative precisely when the series $\sum_{q\in S} \phi(q)\delta_q$ diverges, where $\phi(q)$ denotes Euler's totient function. Otherwise, the set of ``approximable'' real numbers has null measure. In this talk, I will present recent joint work with James Maynard that settles the conjecture of Duffin and Schaeffer.
Date: Friday, December 06, 2019
Time: 2:00PM
Where: Lunt 102
Contact Person: Ilya Khayutin
Contact email: khayutin@northwestern.edu
Contact Phone:
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