On arithmetic properties of integers with missing digits. I: Distribution in residue classes
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.
scientific article; zbMATH DE number 1207957