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Limit laws of maximal Birkhoff sums for circle rotations via quantum modular forms

Bence Borda

Abstract

In this paper, we show how quantum modular forms naturally arise in the ergodic theory of circle rotations. Working with the classical Birkhoff sum $S_N(α)=\sum_{n=1}^N (\{ n α\}-1/2)$, we prove that the maximum and the minimum as well as certain exponential moments of $S_N(r)$ as functions of $r \in \mathbb{Q}$ satisfy a direct analogue of Zagier's continuity conjecture, originally stated for a quantum invariant of the figure-eight knot. As a corollary, we find the limit distribution of $\max_{0 \le N<M} S_N(α)$ and $\min_{0 \le N<M} S_N(α)$ with a random $α\in [0,1]$.

Limit laws of maximal Birkhoff sums for circle rotations via quantum modular forms

Abstract

In this paper, we show how quantum modular forms naturally arise in the ergodic theory of circle rotations. Working with the classical Birkhoff sum , we prove that the maximum and the minimum as well as certain exponential moments of as functions of satisfy a direct analogue of Zagier's continuity conjecture, originally stated for a quantum invariant of the figure-eight knot. As a corollary, we find the limit distribution of and with a random .
Paper Structure (14 sections, 21 theorems, 192 equations, 4 figures)

This paper contains 14 sections, 21 theorems, 192 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Connections to quantum modular forms
  3. Limit laws
  4. The function $h_p$
  5. Local optimum
  6. Factorization of $J_p$
  7. The matching lemma
  8. Asymptotics of $h_p$
  9. Continuity of $h_p$ at irrationals
  10. One-sided limit of $h_p$ at rationals
  11. Quadratic irrationals
  12. Proof of the limit laws
  13. Random rationals
  14. Random reals

Key Result

Theorem 1

Let $\alpha \sim \mu$ with a Borel probability measure $\mu$ on $[0,1]$ which is absolutely continuous with respect to the Lebesgue measure. Then where $E_M = \frac{3}{4 \pi^2} \log M \log \log M + D_{\infty} \log M$ with some constant $D_{\infty} \in \mathbb{R}$, and $\sigma_M = \frac{3}{8 \pi} \log M$.

Figures (4)

  • Figure 1: The function $\log J_{\infty}(r)=\max_{0\le N<q} S_N(r)$ evaluated at all reduced rationals in $[0,1]$ with denominator at most $150$. The graph of $\log J_p(r)$ with $0<p<\infty$ looks very similar, whereas the graph of $\log J_{-p}(r)=-\log J_p(-r)$ is obtained by reflections.
  • Figure 2: The functions $h_{\pm \infty}(r)$ evaluated at all reduced rationals in $[0,1)$ with denominator at most $150$. The asymptotics $1/(8Tr)$ resp. $-1/(8r)$ in Theorem \ref{['asymptoticstheorem']} give a close fit to the graphs.
  • Figure 3: The functions $h_{\infty}(r)$ and $h_2(r)$ evaluated at all reduced rationals in the interval $[0.37,0.38]$ with denominator at most $600$. At the point $3/8=0.375$ the values are $h_{\infty}(3/8)=1/8$ and $h_2(3/8)=0.650008\ldots$. By Theorem \ref{['onesidedlimittheorem']}, the left-hand limits at $3/8$ are $W_{\infty}(3/8)=5/64=0.078125$ and $W_2(3/8)=0.640180\ldots$. The graphs suggest right-continuity at $3/8$.
  • Figure 4: Subtracting the asymptotics from $h_p(r)$ reveals an interesting self-similar structure. Finite $p$ values yield very similar graphs, but the cases $p=\pm \infty$ look markedly different. The four depicted functions are evaluated at all reduced rationals in $[0,1)$ with denominator at most $150$.

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Remark
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark
  • Theorem 7
  • Theorem 8
  • ...and 32 more