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Approximating real numbers by rational numbers with small denominators

Albert Artiles, University of Washington
Thursday, October 6, 2022 - 12:30pm to 1:30pm
PDL C-401

The purpose of this talk is to explore a technique in the theory of geometry and dynamical systems used to solve and understand some statistical question that might arise from number theory. The main theorem we will prove will be the following:

Theorem (Chen, Haynes 2021). For \$\delta > 0\$, define \$q_{\mathrm{min}}(\cdot, \delta) : [0,1] \to \mathbb{N}\$ by $$q_{\mathrm{min}}(x,\delta) = \min \left\{ q \in \mathbb{N} : \exists \, p \in \mathbb{N} \ \ \mathrm{ s.t. } \ \left| x - \frac{p}{q} \right| < \delta \right\}.$$ Then the following limit exists and is positive: $$\lim_{\delta \to 0} \sqrt{\delta} \int_0^1 q_{\mathrm{min}}(x, \delta) \, dx.$$

To be able to sketch a proof of this theorem we will take a tour through different branches of mathematics.

  1. We will construct the space \$X_2\$ of unimodular lattices in \$\mathbb{R}^2\$.
  2. We will create a measure preserving system on \$X_2\$.
  3. Step two will be followed by an equidistribution theorem.
  4. Link Chen and Haynes’s function to the theory of lattices.
  5. Complete the proof.