Distribution of integers with digit restrictions via Markov chains

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.

💡 Research Summary

The paper introduces a novel probabilistic framework for studying the distribution of integers that satisfy digit‑wise restrictions, such as missing‑digit sets and more general multiplicatively invariant sets. The authors replace the traditional Fourier‑analytic approach with a Markov‑chain construction that captures the evolution of residue classes as digits are appended.

For a fixed base g≥2 and a digit set D⊂{0,…,g−1}, the missing‑digit set C_{g,D} consists of all non‑negative integers whose base‑g expansion uses only digits from D. This set is a special case of a ×g‑invariant (multiplicatively invariant) set: any such set is closed under deleting the most significant digit and under deleting the least significant digit in base g. The paper studies the asymptotic proportion

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