arXiv subjects

Léo Vivion

Publications and source records attributed to Léo Vivion.

3 recordsLinked to original sources

arXiv2026

A proof of Rauzy's conjecture on abelian complexity

We resolve a conjecture posed by Rauzy in 1983 concerning the $d$-ary generalizations of Sturmian words. A classical theorem by Coven and Hedlund from 1973 states that Sturmian words, which are classically defined by their subword complexity (a combinatorial refinement of topological entropy), or as the natural codings of irrational rotations on the circle, are also characterized by their abelian complexity (a combinatorial refinement of the notion of discrepancy). More precisely, Sturmian words are exactly the binary infinite words with rationally independent letter frequencies and minimal abelian complexity, which is in this case constant and equal to $2$. We prove that there exist no infinite ternary words with rationally independent letter frequencies and constant abelian complexity equal to $3$, thereby establishing Rauzy's conjecture.

math.CO↗

arXiv2026

A Normality Conjecture on Rational Base Number Systems

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 deviation from normality 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).

math.NT↗

arXiv2026

New examples of words for which binomial complexities and subword complexity coincide

The complexity of an infinite word can be measured in several ways, the two most common measures being the subword complexity and the abelian complexity. In 2015, Rigo and Salimov introduced a family of intermediate complexities indexed by $k\in\mathbb{N}_{>0}$: the $k$-binomial complexities. These complexities scale up from the abelian complexity, with which the $1$-binomial complexity coincides, to the subword complexity, to which they converge pointwise as $k$ tends to $+\infty$. In this article, we provide four classes of $d$-ary infinite words -- namely, $d$-ary $1$-balanced words, words with subword complexity $n\in\mathbb{N}_{>0}\mapsto n+(d-1)$ (which form a subclass of the so-called quasi-Sturmian words), hypercubic billiard words, and words constructed by repeated Sturmian colorings -- for which this scale ``collapses'', that is, all $k$-binomial complexities, for $k\geq 2$, coincide with the subword complexity. This work generalizes a result of Rigo and Salimov, established in their seminal paper from 2015, which asserts that the $k$-binomial complexity of any Sturmian word coincides with its subword complexity whenever $k\geq 2$.

math.CO↗