On the Duffin-Schaeffer conjecture

On the Duffin-Schaeffer conjecture

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Dimitris Koukoulopoulos, Université de Montréal
Fine Hall 314

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.