Fourier series and Diophantine approximation

Han Yu

Published 2017-04-15Version 1

The Duffin-Schaeffer conjecture says that if $\sum_{p}f(p)\phi(p)/p=\infty$, then for Lebesgue a.e number $x\in [0,1]$, the inequality $|x-q/p|<f(p)/p$ holds for infinitely many coprime pairs $(p,q)$. In this paper, we introduce a Fourier analytic method to deal with this conjecture with extra logarithmic divergence, namely: \[ \frac{f(p)}{\log ^C p}=\infty. \] We obtain a partial result from this extra divergence property which says that \ds holds if we have the above divergence and the inequality $f(p)p>\log ^B p$ either never holds or holds for sufficiently many $p$ in a precise sense. Here the dependence of numbers $C,B$ is that: $ C>(B+5)/2. $