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As usual I expect to be critisised for "duplicating" this question. But I do not! As Gjergji immediately notified, that question was from numerology. The one I ask you here (after putting it in my ...

Let $S \subset \mathbb{R}$ be the set of all real numbers $x$ for which there are infinitely many pairs of prime numbers $p$ and $q$ such that $$ \left|x-\frac{p}{q}\right| < \frac{1}{q^2}. $$ Do ...

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...