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Random walks and quadratic number fields

Bence Borda

TL;DR

The work uncovers a novel bridge between random walks on the circle and the arithmetic of real quadratic fields. By studying S_n α mod 1 with α a quadratic irrational, Borda shows that convergence to uniformity in the quadratic Wasserstein metric W_{T,2} is governed by deep invariants of the associated real quadratic field, including fundamental units and special zeta values of Z-modules. The paper develops a two-pronged approach: a general badly approximable regime giving precise log-N rate constants via Fourier-analytic sums, and a quadratic-irrational regime where these constants are expressed explicitly through Dedekind zeta values and Zagier-type formulas for all quadratic irrationals. This synthesis connects probabilistic limit theorems for circle rotations to classical theories of binary quadratic forms, Pell equations, and zeta-valued constants, and provides concrete computational methods for c1(α) and c2(α).

Abstract

We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group $\mathbb{R}/\mathbb{Z}$ in which each step is a random integer multiple of a given quadratic irrational $α$, we show that the rate of convergence to uniformity in the quadratic Wasserstein metric (also known as the periodic $L^2$ discrepancy) is governed by deep arithmetic invariants of the ring of algebraic integers of the real quadratic field $\mathbb{Q}(α)$, such as fundamental units and special values of zeta functions.

Random walks and quadratic number fields

TL;DR

The work uncovers a novel bridge between random walks on the circle and the arithmetic of real quadratic fields. By studying S_n α mod 1 with α a quadratic irrational, Borda shows that convergence to uniformity in the quadratic Wasserstein metric W_{T,2} is governed by deep invariants of the associated real quadratic field, including fundamental units and special zeta values of Z-modules. The paper develops a two-pronged approach: a general badly approximable regime giving precise log-N rate constants via Fourier-analytic sums, and a quadratic-irrational regime where these constants are expressed explicitly through Dedekind zeta values and Zagier-type formulas for all quadratic irrationals. This synthesis connects probabilistic limit theorems for circle rotations to classical theories of binary quadratic forms, Pell equations, and zeta-valued constants, and provides concrete computational methods for c1(α) and c2(α).

Abstract

We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group in which each step is a random integer multiple of a given quadratic irrational , we show that the rate of convergence to uniformity in the quadratic Wasserstein metric (also known as the periodic discrepancy) is governed by deep arithmetic invariants of the ring of algebraic integers of the real quadratic field , such as fundamental units and special values of zeta functions.

Paper Structure

This paper contains 23 sections, 14 theorems, 208 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Context
  3. Notation
  4. Diophantine approximation and continued fractions
  5. Probability metrics
  6. General random walks on the circle group
  7. Lattice random walks on the circle group
  8. Random walks with badly approximable numbers
  9. Diophantine sum estimates
  10. The moments of an exponential sum
  11. Proof of Theorem \ref{['badlyapproxtheorem']}
  12. Proof of Corollaries \ref{['badlyapproxcorollary']} and \ref{['noasymptoticcorollary']}
  13. Random walks with quadratic irrationals
  14. Binary quadratic forms
  15. Diophantine sums with quadratic irrationals
  16. ...and 8 more sections

Key Result

Theorem 1

Assume $\mathbb{E} (X_1) =0$ and $0<\mathbb{E} (X_1^2) < \infty$, and $\alpha$ is a quadratic irrational. Then with $L = \mathrm{gcd} (\mathrm{supp} (X_1))$, $\sigma^2 = \mathbb{E} (X_1^2)$ and explicit positive constants $c_1(L \alpha)$, $c_2(L \alpha)$. If in addition $\mathbb{E} (|X_1|^p) <\infty$ with some real constant $p>2$, then with implied constants depending only on $\alpha$ and the di

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 17 more