2
$\begingroup$

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?

  • 0
    Apologies, 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
  • 0
    Another question: which one of Kronecker's theorems?2012-11-24

2 Answers 2