arXiv subjects

Mélodie Andrieu

Publications and source records attributed to Mélodie Andrieu.

5 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

Infinite Words with very Low Factor Complexity: an introduction to Combinatorics on Words

These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following guiding questions: What is the minimal complexity of a non-trivial infinite word over a binary, ternary, or more generally finite alphabet? How should ''non-triviality'' be formalized? Which words achieve this minimal complexity? Are there many? Are they interesting? In exploring these questions, we introduce classical objects and tools from combinatorics on words -- such as Sturmian words and Rauzy graphs -- as well as little-known and new results. In particular, the third chapter is devoted to a theorem by R. Tijdeman from 1999, which generalizes a seminal result of M. Morse and G. Hedlund from 1938. We provide a new, algebraic proof of this theorem (due to J. Cassaigne and the author, 2022) and develop its consequences.

math.CO↗

arXiv2021

A Rauzy fractal unbounded in all directions of the plane

We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to $\mathbb{Z}^3$. In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.

math.DS↗

arXiv2019

Morphic words and equidistributed sequences

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $ν_w=(ν[n])_n$, equidistributed on $[0,1]$ and such that $ν[m]<ν[n]$ if and only if $σ^m(w)<σ^n(w)$, where $σ$ is the shift operation, erasing the first symbol of $w$. The sequence $ν_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $ν_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $ν_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $φ$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $φ$.

math.DS↗