| Communications in Mathematics |
Editors: Emilie Charlier, Julien Leroy & Michel Rigo
Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $\beta >1$ and derive a necessary condition for validity of the Midy property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property.
We revisit and generalize inequalities for the summatory function of the sum of digits in a given integer base. We prove that several known results can be deduced from a theorem in a 2023 paper by Mohanty, Greenbury, Sarkany, Narayanan, Dingle, Ahnert, and Louis, whose primary scope is the maximum mutational robustness in genotype-phenotype maps.
We revisit a classic theorem of Frougny and Sakarovitch concerning automata for $\varphi$-representations, and show how to obtain it in a different and more computationally direct way. Using it, we can find simple, induction-free proofs of existing results in the literature about these representations, in a uniform and straightforward manner. In particular, we can easily and "automatically'' recover many of the results of recent papers of Dekking and Van Loon. We also obtain a number of new results on $\varphi$-representations.
Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which generate topological spaces. Simple examples show the potential of the concept. Spaces generated by automata are topologically self-similar. Two basic algorithms are outlined. The first one determines automata for all $k$-tuples of equivalent addresses from the automaton for double addresses. The second one constructs finite topological spaces which approximate the generated space. Finally, we discuss the realization of automata-generated spaces as self-similar sets.
We study various aspects of Dyck words appearing in binary sequences, where $0$ is treated as a left parenthesis and $1$ as a right parenthesis. We show that binary words that are $7/3$-power-free have bounded nesting level, but this no longer holds for larger repetition exponents. We give an explicit characterization of the factors of the Thue-Morse word that are Dyck, and show how to count them. We also prove tight upper and lower bounds on $f(n)$, the number of Dyck factors of Thue-Morse of length $2n$.
Conway and Ryba considered a table of bi-infinite Fibonacci sequences and discovered new interesting patterns. We extend their considerations to tables that are defined by the recurrence $X_{n+1}=dX_n+X_{n-1}$ for natural numbers $d$. In our search for new patterns we run into a Red Wall and exotic numeration systems.