A Normality Conjecture on Rational Base Number Systems
Mélodie Andrieu, Shalom Eliahou, Léo Vivion
TL;DR
This paper posits a strong normality conjecture for rational base number systems, asserting that every minimal word over the alphabet $\{0,\dots,q-1\}$ and every maximal word over $\{p-q,\dots,p-1\}$ is normal. It establishes the equivalence between normality and equidistribution of the $T_{p/q}$ iterates, and shows that normality would imply the nonexistence of $Z_{p/q}$-numbers (a Mahler-type conjecture) and would resolve Akiyama’s triple-expansion conjecture as well as the Dubickas–Mossinghoff $4/3$ problem, among others. The authors provide an explicit algorithm to compute minimal and maximal words and perform extensive numerical experiments across several bases to gather evidence for normality, including analysis of richness thresholds and subword uniformity. The results suggest that minimal words behave like random $q$-ary words in terms of subword distribution and complexity, offering new insights into the structure of rational-base representations and guiding future work on longstanding open questions.
Abstract
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, we present extensive numerical experiments that examine the richness threshold and the discrepancy of these words. We also discuss the implications that the validity of our conjecture would have for several long-standing open problems, including the existence of $Z$-numbers (Mahler, 1968) and $Z_{p/q}$-numbers (Flatto, 1992), the existence of triple expansions in rational base $p/q$ (Akiyama, 2008), and the Collatz-inspired `4/3 problem' (Dubickas and Mossinghoff, 2009).