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Distribution of integers with digit restrictions via Markov chains

Vicente Saavedra-Araya

TL;DR

This work studies how digit-restricted and multiplicatively invariant sets of integers distribute across residue classes, introducing a Fourier-free Markov-chain approach in base $g$. The authors derive a main theorem giving a necessary and sufficient gcd condition for joint uniformity of missing-digits sets in residue classes, and extend the method to the broader class of multiplicatively invariant sets represented by subshifts, via the Fischer cover and follower sets. They also establish results on transversality with arithmetic progressions, showing a clean dichotomy (0 or full dimension) for transitive sofic and, more generally, entropy-minimal subshifts, while presenting counterexamples outside this regime. The framework unifies several classical results (EMS, Gelfond, GMR) under a Markov-chain perspective and yields both qualitative and quantitative convergence statements, including rates in many cases. Overall, the paper provides a versatile, Fourier-free method to analyze uniform distribution questions for fractal-like integer sets and connects these distributional properties to topological entropy and subshift structure.

Abstract

In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach utilizes Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

Distribution of integers with digit restrictions via Markov chains

TL;DR

This work studies how digit-restricted and multiplicatively invariant sets of integers distribute across residue classes, introducing a Fourier-free Markov-chain approach in base . The authors derive a main theorem giving a necessary and sufficient gcd condition for joint uniformity of missing-digits sets in residue classes, and extend the method to the broader class of multiplicatively invariant sets represented by subshifts, via the Fischer cover and follower sets. They also establish results on transversality with arithmetic progressions, showing a clean dichotomy (0 or full dimension) for transitive sofic and, more generally, entropy-minimal subshifts, while presenting counterexamples outside this regime. The framework unifies several classical results (EMS, Gelfond, GMR) under a Markov-chain perspective and yields both qualitative and quantitative convergence statements, including rates in many cases. Overall, the paper provides a versatile, Fourier-free method to analyze uniform distribution questions for fractal-like integer sets and connects these distributional properties to topological entropy and subshift structure.

Abstract

In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach utilizes Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

Paper Structure

This paper contains 20 sections, 37 theorems, 204 equations, 2 figures.

Table of Contents

  1. Introduction
  2. History and context
  3. Overview of main results
  4. Outline of the proof
  5. Preliminaries
  6. Markov chains
  7. Elements of symbolic dynamics
  8. Fractal sets of integers
  9. Construction of Markov chains
  10. Transition matrix
  11. Initial distribution
  12. From Markov chains to uniform distribution
  13. Applications to the distribution of multiplicatively invariant sets of integers
  14. Uniform distribution of integers with missing digits
  15. Revisiting Gelfond's theorem: An equivalent condition
  16. ...and 5 more sections

Key Result

Corollary 1.2

Let $a,a'\in {\mathbb N}$ such that $\gcd(g-1,a')=1$. Then, for any $b\in {\mathbb Z}_a$ and $b'\in {\mathbb Z}_{a'}$.

Figures (2)

  • Figure 1: Graph that represents the 2-regular and transitive shift of finite type associated to the Matrix $T$ in \ref{['Example2']}.
  • Figure 2: Fischer cover of the sofic shift $\Sigma$ presented in \ref{['ExampleSofic']}.

Theorems & Definitions (98)

  • Definition 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem A
  • Corollary A
  • Theorem B
  • Theorem C
  • Conjecture
  • Theorem 2.1
  • ...and 88 more