On arithmetic properties of integers with missing digits. I: Distribution in residue classes
type: Item
instance of: scholarly article
title: On arithmetic properties of integers with missing digits. I: Distribution in residue classes
zbMATH Open document ID: 0923.11024
author: Christian Mauduit
author: András Sárközy
author: P. Erdős
published in: Journal of Number Theory
publication date: 1999-06-03T00:00:00Z
review text: For an integer \(g\geq 3\) let \(\mathcal D\) be a subset of \(\{0, 1,\dots, g-1\}\) with \(0\in \mathcal D\), and let \(\mathcal W\) be the set of those integers whose development in base \(g\) contains only digits from \(\mathcal D\). The paper studies arithmetic properties of the set \(\mathcal W\). Theorem 1 shows that \(\mathcal W\) is uniformly distributed modulo \(m\) for every integer satisfying \((g(g-1),m)=1\), and gives a sharp estimate for the remainder term. Theorems 2 and 3 consider the question whether \([1, N] \cap \mathcal W\) meets every residue class modulo \(m\) for values of \(m\) that are too large for an application of Theorem 1 (comparable to a power of \(N\)). Theorem 4 considers the distribution of almost \(z\)th power-free integers in \(\mathcal W\) (almost means that a divisibility by \(p^z\), \(p \mid g(g-1)\) is ignored). This problem is not completely settled, and the authors mention the unsolved problem whether for \(g\geq 6\) there are infinitely many squarefree numbers containing only the digits \(0,1\). Finally Theorem 5 exhibits elements of \(\mathcal W\) that are divisible by the \(z\)th power of a large prime.
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 1207957
zbMATH Keywords: digits
zbMATH Keywords: number systems
zbMATH Keywords: uniform distribution
reviewed by: Imre Z. Ruzsa
Wikidata QID: Q102321444
MaRDI profile type: Publication
full work available at URL: jnth.1998.2229
OpenAlex ID: W2003176801
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DOI: 10.1006/JNTH.1998.2229
description: scientific article; zbMATH DE number 1207957