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A statistical model for points expanding in higher dimensions while being tied to bijective involutions

Cristian Cobeli, The Nguyen, Alexandru Zaharescu

Abstract

Let $\mathcal{M}$ be a set with $M$ elements, let $ψ:\mathcal{M}\to\mathcal{M}$ be a bijective involution, and let~$\boldsymbol{\mathcal{X}}_ψ$ be the set of sequences $(x_1,\dots,x_M)\in\mathcal{M}^M$ with the property that $x_{M+1-j} = ψ(x_j)$ for $1\le j\le M$. This framework can be used to infer the possible distribution of sequences, such as the modular ones, that pose challenges for conventional methods. We prove that when $M$ is even, there exists a limit probability density function that weighs the parameter $k$ that counts the appearances of the elements of $\mathcal{M}$ among the terms of sequences $\textbf{x}\in\boldsymbol{\mathcal{X}}_ψ$. It turns out that the number of fixed points of $ψ$ influences the probability density function, which decomposes into two pieces, each multiplied by complementary factors, and the smaller of the two pieces appears only when $k$ is even. Applying the model, we find a threshold from which almost all sequences contain related terms with prescribed frequencies.

A statistical model for points expanding in higher dimensions while being tied to bijective involutions

Abstract

Let be a set with elements, let be a bijective involution, and let~ be the set of sequences with the property that for . This framework can be used to infer the possible distribution of sequences, such as the modular ones, that pose challenges for conventional methods. We prove that when is even, there exists a limit probability density function that weighs the parameter that counts the appearances of the elements of among the terms of sequences . It turns out that the number of fixed points of influences the probability density function, which decomposes into two pieces, each multiplied by complementary factors, and the smaller of the two pieces appears only when is even. Applying the model, we find a threshold from which almost all sequences contain related terms with prescribed frequencies.
Paper Structure (17 sections, 9 theorems, 92 equations)

This paper contains 17 sections, 9 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Assumptions of the model
  3. The main results
  4. Notation
  5. The average size of the ratios mk(x) over M
  6. The second moment about the mean
  7. The case k is odd
  8. The case k even
  9. Estimating V1(k,M)
  10. Estimating V2(k,M)
  11. Estimating V3(k,M)
  12. Ending the estimation of the second moment when k is even
  13. Proof of Theorem \ref{['Theorem1']}
  14. Proof of Theorem \ref{['Theorem2']} and \ref{['Theorem3']}
  15. Proof of Theorem \ref{['Theorem2']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be a set of cardinality $M$, where $M$ is a positive even integer. Let $\psi=\psi_{\mathcal{M}}$ be an involutive permutation of $\mathcal{M}$ and consider the set ${\bm\textbf{\mathcal{X}}}_{\!\psi}$ of $\psi$-symmetric vectors of length $M$ defined by eqDefinitionbXM. Suppose the

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • Lemma 3.2
  • Remark 4.1
  • Lemma 4.1
  • ...and 3 more