Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the records and zeros of a deterministic random walk

Henk Bruin, Robbert Fokkink

TL;DR

This work analyzes the records and zeros of a deterministic random walk S_n(ξ) induced by irrational rotations, focusing on ξ = 2sqrt 2 and its relation to the OEIS sequences A120243 and A120749. Using Pell numbers, Ostrowski representations, and automata theory, the authors prove that the records for S_n(2sqrt 2) are the half-Pell numbers and are Pell-automatic, while the zeros admit a Pell-automatic description; they also show that the records for S_n(sqrt 2) satisfy a Pell-like recurrence and are Pell-automatic. Kimberling's questions are settled in the BR-number regime, and the difference b(n)-a(n) is shown to take every positive value infinitely often, via symmetry arguments and discrepancy theory; these results extend to noble mean rotations, revealing a deep connection between deterministic walks, automatic sequences, and linear recurrences. The authors conjecture that the records of deterministic random walks are ξ-Ostrowski automatic for quadratic irrationals, suggesting a broad automata-theoretic structure underlying these dynamical systems with potential implications for number representations and symmetry in rotation dynamics.

Abstract

We settle two questions on sequence A120243 in the OEIS that were raised by Clark Kimberling and partly solve a conjecture of Van de Lune and Arias de Reyna. We extend Kimberling's questions to the framework of deterministic random walks, automatic sequences, and linear recurrences. Our results indicate that there may be a deeper connection between these structures. In particular, we conjecture that the records of deterministic random walks are $ξ$-Ostrowski automatic for a quadratic rotation number $ξ$.

On the records and zeros of a deterministic random walk

TL;DR

This work analyzes the records and zeros of a deterministic random walk S_n(ξ) induced by irrational rotations, focusing on ξ = 2sqrt 2 and its relation to the OEIS sequences A120243 and A120749. Using Pell numbers, Ostrowski representations, and automata theory, the authors prove that the records for S_n(2sqrt 2) are the half-Pell numbers and are Pell-automatic, while the zeros admit a Pell-automatic description; they also show that the records for S_n(sqrt 2) satisfy a Pell-like recurrence and are Pell-automatic. Kimberling's questions are settled in the BR-number regime, and the difference b(n)-a(n) is shown to take every positive value infinitely often, via symmetry arguments and discrepancy theory; these results extend to noble mean rotations, revealing a deep connection between deterministic walks, automatic sequences, and linear recurrences. The authors conjecture that the records of deterministic random walks are ξ-Ostrowski automatic for quadratic irrationals, suggesting a broad automata-theoretic structure underlying these dynamical systems with potential implications for number representations and symmetry in rotation dynamics.

Abstract

We settle two questions on sequence A120243 in the OEIS that were raised by Clark Kimberling and partly solve a conjecture of Van de Lune and Arias de Reyna. We extend Kimberling's questions to the framework of deterministic random walks, automatic sequences, and linear recurrences. Our results indicate that there may be a deeper connection between these structures. In particular, we conjecture that the records of deterministic random walks are -Ostrowski automatic for a quadratic rotation number .

Paper Structure

This paper contains 7 sections, 18 theorems, 35 equations, 5 figures, 1 table.

Table of Contents

  1. Irrational rotations and automata
  2. The records of $S_n(2\sqrt 2)$
  3. The zeros of $S_n(2\sqrt 2)$
  4. The difference sequence $b-a$
  5. Self-similarity of noble mean rotations
  6. Concluding questions and remarks
  7. Acknowledgement

Key Result

Theorem 1

Let $I=[0,\frac{h}{k})$ and let $\frac{p_n}{q_n}$ be the convergents of $\xi$. The discrepancy $N(\xi,I)-\ell(I)\cdot N$ is nonnegative if and only if $k\ |\ q_{2n+1}$ for all $n$.

Figures (5)

  • Figure 1: A finite state automaton that decides if a number $n$ is in A120749 or in its complement A120243. The input is in msd format. Inputs that end in double circled states (acceptance) are in A120749. Inputs that end in state 1 (rejection) are in A120243. Inputs that end in $2$ are not a valid Pell representation. The automaton is produced by the automatic theorem prover Walnut using the results in schaeffer2024.
  • Figure 2: An automaton that accepts the records of $S_n(\sqrt 2)$ in Pell numeration in msd representation.
  • Figure 3: An automaton that accepts the records of $S_n(2\sqrt 2)$ in Pell numeration in msd representation.
  • Figure 4: An automaton that accepts the records of $S_n(\sqrt 3)$ in $\sqrt 3/2$-Ostrowski numeration, provided that the system of recurrences conjectured by Van de Lune and Arias de Reyna holds. The continued fraction of $\sqrt{3}/2$ has partial coefficients $[0;1,\overline{6,2}]$, where the bar marks that these coefficients repeat. The denominators of its convergents are $1, 7, 15, 97, \ldots$.
  • Figure 5: An automaton that accepts the zeros of $S_n(2\sqrt 2)$ in Pell numeration in msd representation.

Theorems & Definitions (34)

  • Theorem 1: Boshernitzan and Ralston
  • Corollary 2
  • Theorem 3: Van de Lune
  • Corollary 4
  • proof
  • Theorem 5
  • proof
  • Definition 6
  • Conjecture 7
  • Theorem 8
  • ...and 24 more