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Democratic sequences

Anne Bertrand Mathis

TL;DR

The paper addresses realizing prescribed letter frequencies on finite or countable alphabets by constructing a democratic sequence that iteratively selects the most underrepresented symbol. A deficit-excess framework and a tie-breaking rule guarantee that each symbol $a$ appears with the prescribed frequency $\lambda(a)$, with Theorem 1 providing the convergence $\frac{1}{N}\#\{n\le N: u_n=a\} \to \lambda(a)$, even for countable alphabets and with a Joker of zero frequency. It then connects this combinatorial construction to dynamical and numeration contexts: (i) any probability measure on the torus can be approximated by measures associated with sequences of fractional parts $\{\beta^n\}$ modulo one, yielding a dense set of associated measures under weak convergence (Theorem 2); and (ii) a base-$10$ numeration example shows how the democratic process induces a digit-growth order and links to automatic sequences via a morphic fixed point. Collectively, the results bridge symbolic frequency realization, dynamical-measure approximation, and numeration theory, revealing new ways to realize arbitrary distributions through explicit, deterministic constructions.

Abstract

Given a countable alphabet and a sountable set pf preset frequency, we construct a sequence where each letter appears with the preaasigned frequency.

Democratic sequences

TL;DR

The paper addresses realizing prescribed letter frequencies on finite or countable alphabets by constructing a democratic sequence that iteratively selects the most underrepresented symbol. A deficit-excess framework and a tie-breaking rule guarantee that each symbol appears with the prescribed frequency , with Theorem 1 providing the convergence , even for countable alphabets and with a Joker of zero frequency. It then connects this combinatorial construction to dynamical and numeration contexts: (i) any probability measure on the torus can be approximated by measures associated with sequences of fractional parts modulo one, yielding a dense set of associated measures under weak convergence (Theorem 2); and (ii) a base- numeration example shows how the democratic process induces a digit-growth order and links to automatic sequences via a morphic fixed point. Collectively, the results bridge symbolic frequency realization, dynamical-measure approximation, and numeration theory, revealing new ways to realize arbitrary distributions through explicit, deterministic constructions.

Abstract

Given a countable alphabet and a sountable set pf preset frequency, we construct a sequence where each letter appears with the preaasigned frequency.
Paper Structure (4 sections, 6 equations)

This paper contains 4 sections, 6 equations.

Table of Contents

  1. Presentation of the method.
  2. Examples and applications.
  3. Approximation of measures on the torus $\mathbb{R}/\mathbb{Z}$ by measures associated with sequences $(\beta^{n})_{n\geq1}$ modulo one.
  4. Numeration.