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The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: http://en.wikipedia.org/wiki/...

I am looking for an upper bound $R=R_{n,\varepsilon}$ such that for given $\varepsilon>0$ and real numbers $\alpha_1, \dotsc, \alpha_n$ in, say, $[1,2]$, there is $x\in [1,R]$ such that $$ \frac 1n\...

This problem could be viewed as a polynomial generalization of the Lonely runner conjecture. And $p$, $n$ are taken sufficiently large. Take $n\in \mathbb{N}^*$ fixed, $A_p \subset (\mathbb{Z} / p \...

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...