[Paper Review] Statistical properties of the Calkin--Wilf tree: real an p-adic distribution

Giedrius Alkauskas, Jörn Steuding|ArXiv.org|Dec 29, 2007

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper investigates the statistical distribution of rational numbers in the Calkin–Wilf tree, proving that the mean value of the $ n $th generation is $ \Sigma(n) = 3 \cdot 2^{n-2} - \frac{1}{2} $, converging to $ 1.5 $ as $ n \to \infty $, and establishes a $ p $-adic analog of this distribution using Markov chains on $ p $-adic integers, showing finite orbit structures and recurrence in the associated transition matrices.

ABSTRACT

We examine statistical properties of the Calkin--Wilf tree and give number-theoretical applications.

Motivation & Objective

• To analyze the statistical distribution of rational numbers in the Calkin–Wilf tree, particularly the mean value of elements in each generation.
• To extend the real-valued statistical properties of the Calkin–Wilf tree to the $ p $-adic setting, establishing a $ p $-adic analog of the mean-value distribution.
• To model the distribution using Markov chains on $ p $-adic integers, characterizing recurrence and finite orbit structures in the state space.
• To prove that the $ p $-adic distribution of the tree's elements is governed by a finite, irreducible, and aperiodic Markov chain with explicit recurrence properties.

Proposed method

• Uses induction to prove the closed-form expression for the sum $ \Sigma(n) = 3 \cdot 2^{n-2} - \frac{1}{2} $ of the $ n $th generation of the Calkin–Wilf tree.
• Employs symmetry properties of the tree: $ x_j^{(n)} = \frac{a}{b} \iff x_{2^{n-1}+1-j}^{(n)} = \frac{b}{a} $, to pair elements and simplify summation.
• Defines $ p $-adic valuation-based states $ (i, \kappa) $ and constructs transition maps $ \tau(i,\kappa) $ and $ \sigma(i,\kappa) $ to model the tree's evolution in the $ p $-adic setting.
• Constructs an infinite transition matrix $ \mathcal{A} $ for the Markov chain, decomposed into finite, irreducible, and aperiodic blocks $ \mathbf{P}_\kappa $ of size $ p^\kappa + p^{\kappa-1} $.
• Proves that each orbit under the transition maps is finite and recurrent, with all states communicating, ensuring convergence to a unique stationary distribution.
• Uses valuation arguments to show that $ v\left(\frac{a}{a+b} - i\right) \geq \kappa $ if and only if $ v\left(\frac{a}{b} - i_0\right) \geq \kappa_0 $, linking real and $ p $-adic distributions.

Experimental results

Research questions

• RQ1 What is the exact mean value of the elements in the $ n $th generation of the Calkin–Wilf tree?
• RQ2 How does the distribution of rational numbers in the Calkin–Wilf tree behave under $ p $-adic valuation?
• RQ3 Can the statistical properties of the Calkin–Wilf tree be extended to the $ p $-adic numbers via a Markov chain model?
• RQ4 What is the structure of the state space and transition dynamics in the $ p $-adic version of the Calkin–Wilf tree?
• RQ5 Are the $ p $-adic transition matrices irreducible and aperiodic, ensuring convergence to a unique stationary distribution?

Key findings

• The sum of all elements in the $ n $th generation of the Calkin–Wilf tree is exactly $ \Sigma(n) = 3 \cdot 2^{n-2} - \frac{1}{2} $, implying the mean value converges to $ \frac{3}{2} $ as $ n \to \infty $.
• The $ p $-adic distribution of the Calkin–Wilf tree is modeled by a Markov chain with finite, irreducible, and aperiodic transition matrices $ \mathbf{P}_\kappa $ of size $ p^\kappa + p^{\kappa-1} $.
• Each $ p $-adic orbit under the transition maps has exactly $ p^\kappa + p^{\kappa-1} $ distinct states, forming a finite, recurrent class.
• The transition matrix $ \mathcal{A} $ is block-diagonal with irreducible blocks $ \mathbf{P}_\kappa $, each corresponding to a unique $ \kappa \geq 1 $, ensuring long-term statistical stability.
• The $ p $-adic analog of the mean-value property is preserved: the stationary distribution on each orbit is uniform, and the system exhibits recurrence and communication between all states.
• The $ p $-adic transition rules $ \tau(i,\kappa) $ and $ \sigma(i,\kappa) $ are defined via valuation-based transformations, ensuring compatibility with the tree's recursive structure.