Numbers omitting digits in certain base expansions
Alexia Yavicoli, Han Yu
TL;DR
The paper addresses the problem of finding integers whose representations in several bases simultaneously omit prescribed digits, providing explicit quantitative thresholds. It uses a thickness-based fractal framework and the Fal-Yav intersection theorem to guarantee nonempty intersections of thick digit-omitting sets, yielding infinite families of such integers; the key quantitative device is the threshold $M(k)$ defined by $g_k(M-2)=\frac{M-2}{\log(M-2)}-k \frac{e 4 (432)^2}{\log(4)}>0$. The work includes a zero-digit case, extensions to nonzero digit sets, and an algebraic extension to bases in number fields (including $\mathbb{Q}[i]$), demonstrating that for large enough bases and equal-norm embeddings there are infinitely many integers (or algebraic integers) whose expansions avoid a fixed digit in all bases. This blends fractal geometry of missing-digit sets with number-theoretic base expansions, producing constructive, quantitative results and broadening the scope to algebraic bases with potential for further generalization.
Abstract
In DOI:10.1017/etds.2022.2 the author proved that for each integer $k$ there is an implicit number $M > 0$ such that if $b_1, \cdots , b_k$ are multiplicatively independent integers greater than $M$, there are infinitely many integers whose base $b_1, b_2, \cdots , b_k$ expansions all do not have zero digits. In this paper we don't require the multiplicative independence condition and make the result quantitative, getting an explicit value for $M$. We also obtain a result for the case when the missing digit(s) may not be zero. Finally, we extend our method to study various missing-digit sets in an algebraic setting.