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Prescribed distinct-digit growth in countable alphabets

Ying Wai Lee

TL;DR

The paper analyzes the growth of distinct symbols $D_n$ in digit expansions generated by affine full-branch countable IFS with regularly varying branch weights. It proves a sharp phase transition in the Hausdorff dimension of exceptional growth sets: any positive linear growth yields $ ext{dim }E_ heta=1/ ho$, while a broad class of sublinear growth rates preserves full dimension $ ext{dim }E_f=1$, with the Lüroth case yielding $ ext{dim }E_ heta=1/2$. The results are obtained by a probabilistic-occupancy framework combined with fractal-geometry tools, including mass distribution, Hausdorff–Cantelli, and tilting arguments exploiting regular variation. These findings illuminate the fractal geometry of atypical distinct-digit growth in heavy-tailed digit systems and extend classical results from continued fractions and Lüroth expansions to a unified, i.i.d. digit setting.

Abstract

The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in which the distinct-digit count grows at a positive linear rate or at a prescribed sublinear rate are determined. The resulting dimension laws exhibit a sharp phase transition: imposing any positive linear rate forces the dimension to collapse to a value determined solely by the tail index, whereas under a broad class of sublinear growth rates, the exceptional sets retain full Hausdorff dimension.

Prescribed distinct-digit growth in countable alphabets

TL;DR

The paper analyzes the growth of distinct symbols in digit expansions generated by affine full-branch countable IFS with regularly varying branch weights. It proves a sharp phase transition in the Hausdorff dimension of exceptional growth sets: any positive linear growth yields , while a broad class of sublinear growth rates preserves full dimension , with the Lüroth case yielding . The results are obtained by a probabilistic-occupancy framework combined with fractal-geometry tools, including mass distribution, Hausdorff–Cantelli, and tilting arguments exploiting regular variation. These findings illuminate the fractal geometry of atypical distinct-digit growth in heavy-tailed digit systems and extend classical results from continued fractions and Lüroth expansions to a unified, i.i.d. digit setting.

Abstract

The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in which the distinct-digit count grows at a positive linear rate or at a prescribed sublinear rate are determined. The resulting dimension laws exhibit a sharp phase transition: imposing any positive linear rate forces the dimension to collapse to a value determined solely by the tail index, whereas under a broad class of sublinear growth rates, the exceptional sets retain full Hausdorff dimension.
Paper Structure (10 sections, 16 theorems, 148 equations, 1 figure)

This paper contains 10 sections, 16 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Continued fractions
  3. Affine full-branch countable iterated function systems
  4. Classical Lüroth expansions
  5. Classical infinite-urn occupancy theory
  6. Main Results
  7. Proof of Theorem \ref{['thm:linear']}
  8. Block concatenation construction
  9. Hausdorff--Cantelli and tilting
  10. Proof of Theorem \ref{['thm:sublinear']}

Key Result

Theorem 1

For Lebesgue almost every $x\in[0,1)\setminus\mathbb{Q}$, as $n\to+\infty$, where the exponent is given by:

Figures (1)

  • Figure 1: Gauss map and Lüroth map

Theorems & Definitions (31)

  • Theorem : Wu--Xie WuXie2017RangeRenewalCF
  • Theorem : Wu--Xie WuXie2017RangeRenewalCF
  • Definition
  • Theorem 1
  • Definition
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof
  • Proposition 1
  • ...and 21 more