Given an irrational number $x \in \mathbb{R}\setminus \mathbb{Q}$, is it possible to find a map $T: \mathbb{N} \to \mathbb{N}$ strictly increasing such that $\left\{T(n) x\right\} \to 0$ as $n \to \infty$, where $\left\{\cdot\right\}$ is the fractional part?
Limit involving fractional part of irrational numbers
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real-analysis
limits
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0Apologies, I realized Dirichlet's theorem is not appropriate here, we must instead use (a one variable version of) Kronecker's theorem to get $qx$ within $1/2N$ of $1/2N$. So the result depends on a slightly stronger theorem but still much weaker than uniform distribution. – 2012-11-24
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0Another question: which one of Kronecker's theorems? – 2012-11-24