On arithmetic properties of integers with missing digits. II: Prime factors
For an integer \(g\geq 3\) let \(\mathcal D\) be a subset of \(\{0, 1, ..., g-1\}\) 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 continues the study of the arithmetic properties of the set \(\mathcal W\), started in part I [J. Number Theory 70, 99-120 (1998; Zbl 0923.11024)]. Write \({\mathcal D } = \{d_1, ..., d_t\}\). Under the assumptions \((d_1, ..., d_t)=1\) and \(t>\sqrt g\), the authors prove a limiting law for the distribution of \(\omega (n)\) for \(n\in \mathcal D\), \(n\leq N\), and show that the maximal order of \(\omega \) is \( \gg ( \log N)/ \log\log N \). They list exciting open problems, like finding odd and even values of \(\omega \) in \(\mathcal D\), or finding almost primes in this set. As a partial result concerning the second problem, they give estimates for the largest prime factor of elements of \(\mathcal D\).
scientific article; zbMATH DE number 1334447
scientific article