Jean-Paul Allouche, Manon Stipulanti
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.
Savinien Kreczman, Sébastien Labbé, Manon Stipulanti
Introduced in 2001 by Lecomte and Rigo, abstract numeration systems provide a way of expressing natural numbers with words from a language $L$ accepted by a finite automaton. As it turns out, these numeration systems are not necessarily positional, i.e., we cannot always find a sequence $U=(U_i)_{i\ge 0}$ of integers such that the value of every word in the language $L$ is determined by the position of its letters and the first few values of $U$. Finding the conditions under which an abstract numeration system is positional seems difficult in general. In this paper, we thus consider this question for a particular sub-family of abstract numeration systems called Dumont--Thomas numeration systems. They are derived from substitutions and were introduced in 1989 by Dumont and Thomas. We exhibit conditions on the underlying substitution so that the corresponding Dumont--Thomas numeration is positional. We first work in the most general setting, then particularize our results to some practical cases. Finally, we link our numeration systems to existing literature, notably properties studied by Rényi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995