Savinien Kreczman

Numbers in square brackets refer to the corresponding publications in the list of publications.

Numeration systems [2,3,5,6,7]

A single number (say, the product of the first, second and fourth primes) can be represented in multiple different ways (42, XLII, 0x2a) depending on which convention is used. However, not all systems are equally suited for all tasks: for instance, multiplication is harder when working with roman numerals than arabic numerals. This double realization leads us to the study of numeration systems, which are conventions for the representation of numbers, that prescribe what word can be used to mean which number. Questions that can be asked about those include "Can we easily recognize a word to be the representation of some number?", "Can we test if a number belongs to a given set, using only its representation ?" or "What is the set of numbers that have periodic expansions?".

Types of numeration systems that I study include:

• Positional numeration systems [3,5]: These are used to represent natural numbers. A sequence U of weights is given, and every natural number is decomposed as a sum of elements of U. The representation of a number n is deduced from this decomposition. Examples include the decimal numeration system, where the sequence U is the sequence of powers of ten, and the Zeckendorf numeration system based on the Fibonacci sequence.

• Rényi numeration systems : These are used to represent nonnegative real numbers. A number is decomposed as a (possibly infinite) sum of powers of some base number β, then the representation of x is deduced from this decomposition. This representation is a right-infinite word. Examples include the usual decimal numeration system, which has the base 10, and the golden base numeration system, where the base is the golden ratio.

• Alternate base numeration systems [2,5,6,7]: A generalization of Rényi numeration systems. Rather than using a single base, the base that is used cycles between one of p possibilities. Their introduction by Charlier and Cisternino was motivated by the study of regularity in positional numeration systems (see [5]).

• Dumont-Thomas numeration systems [3]: Used to represent natural numbers. From a substitution, define an infinite labeled tree and organize it in columns, then represent n by the label of a shortest path from the root to column n. 

There exists a link between positional numeration systems with a dominant root (which is most, but not all of them), Rényi numeration systems, and Dumont-Thomas numeration systems based on substitutions with fixed points. For instance, the Zeckendorf numeration system, the golden base system, and the system based on the Fibonacci substitution have the same language, up to technicalities. If we wish to study positional systems without a dominant root or substitutions without a fixed point, alternate base systems form a suitable generalization of Rényi systems.

Combinatorics on words [1,4]

Words, in the mathematical sense, are sequences of symbols. The accent is put on the syntax here, not the semantics. A DNA sequence is a word on four letters, a sequence of "turn left", "turn right" and "go straight" is a word on three letters. We may then study syntactic properties of these words, such as the amount of different short sequences that can appear in them, or how much repetition is present in the word.

Publications and preprints:

1. Magic numbers in periodic sequences. With Luca Prigioniero, Eric Rowland and Manon Stipulanti. In Combinatorics on Words (LNCS 13899), 206-219 (2023). arXiv (long version), Journal 

Periodic sequences are known to be automatic, regular and recursive. We study their complexity, in terms of the number of states of a corresponding automaton, the dimension of a linear representation, or the order of a recurrence relation.

2. On periodic alternate base expansions. With Émilie Charlier and Célia Cisternino. Journal of Number Theory 254, 184-198 (2024). arXiv, Journal

In 1980, Schmidt studied which Rényi bases β are such that the numbers with periodic expansions are exactly those in Q(β). We generalize his results to the case of alternate bases.

3. A succinct study of positionality for Dumont-Thomas numeration systems. With Manon Stipulanti and Sébastien Labbé. In Combinatorics on Words (LNCS 15729), 179-191 (2025). arXiv (long version), Journal

In her PhD thesis, Lepšová asked to characterize which Dumont-Thomas numeration systems can also be defined as positional. We answer this question with a full criterion. We also show that these positional Dumont-Thomas systems correspond to another known family of numeration systems, the Bertrand systems.

4. On two conjectures of Shallit about Thue-Morse-like sequences. With Ľubomíra Dvořáková and Edita Pelantová. European Journal of Combinatorics 132 B, 104250 (2026). arXiv, Journal

In a 2025 article, Shallit introduced a family of words generalizing the famous Thue-Morse word, and put forth two conjectures on this family, one on their difference of factor complexity and one on their critical exponent. We prove both conjectures by studying the bispecial factors of these words using a method of Klouda.

5. Numeration systems without a dominant root and regularity. With Émilie Charlier. Submitted. arXiv

We study for which positional numeration systems the language of all representations is regular. This prolongs work by Hollander, who in 1998 used a connection between positional systems with a dominant root and Rényi systems to obtain necessary and sufficient conditions. In this article, we explain why alternate bases are the correct generalization of Rényi systems and how they allow us to tackle the case without a dominant root. We provide a characterization of the systems with a regular language and refute a conjecture of Hollander.

6. Parry condition, existence and uniqueness of alternate bases. With Zuzana Masáková and Edita Pelantová. Submitted. arXiv

When dealing with alternate bases, it is common to provide examples by specifying the representations of 1 rather than the bases themselves. However, for this to be possible we must first prove that there indeed exists a base with the given representations of 1. While this is well-understood in the Rényi case, the situation is not so clear for alternate bases. In this article, we give criteria for the existence and uniqueness of bases with given representations of 1.

7. The Maximal Digit Property of numeration systems. With Émilie Charlier, Zuzana Masáková and Edita Pelantová. In preparation.

We generalize to the alternate base case a family of particularly well-behaved Rényi numeration systems, with connections to optimality, confluence, normalization and finiteness properties.

Talks:

Numbers in square brackets indicate the corresponding publication.

1. Alternate bases where all rational numbers have periodic expansions. [2] Journées SDA2 2022, ULiège (Liège, Belgium), June 14, 2022. ORBi

2. On periodic alternate base expansions. [2] Journées Montoises d'Informatique Théorique 2022 (JM2022), Czech Technical University (Prague, Czech Republic), September 5, 2022. ORBi

3. Regularity of the language of greedy numeration systems. [5] Numeration 2023, ULiège (Liège, Belgium), May 23, 2023. ORBi

4. On β-numeration and related objects. Combinatorics on Words 2024, Czech Technical University (Janov nad Nisou, Czech Republic), May 19, 2024. Lecture for undergraduate students.

5. Confluent alternate numeration systems. [7] Journées Montoises d'Informatique Théorique 2024 (JM2024), Université Nice-Côte d'Azur (Nice, France), September 3, 2024. ORBi

6. Periodic Automatic Sequences. [1] HIGHLIGHTS of Logic, Games and Automata 2024, University of Bordeaux (Bordeaux, France), September 19, 2024.

7. Dumont-Thomas numeration systems. [3] PhD Away Days 2024, Luxembourg University (Liège, Belgium), October 21, 2024.

8. Positionality of Dumont-Thomas numeration systems. [3] One World Numeration Seminar (online), May 27, 2025. ORBi

9. A succinct study of positionality for Dumont-Thomas numeration systems. [3] WORDS 2025, LORIA (Nancy, France), July 4, 2025. ORBi

10. The Maximal Digit Property of numeration systems. [7] Numeration and Substitution 2025, University of Tsukuba (Tsukuba, Japan), September 9, 2025. ORBi

11. Factor complexity and critical exponent of words in a Thue-Morse family. [4] One World Combinatorics on Words Seminar (online), January 20, 2026. ORBi

Posters:

Numbers in square brackets indicate the corresponding publication.

1. Régularité de langages générés par un système de numération glouton. [5] Journées Nationales de l'Informatique Mathématique 2023, IRIF (Paris, France), April 4, 2023 (French). ORBi

2. Confluent Alternate Numeration Systems. [7] Numeration 2024, University of Utrecht (Utrecht, Netherlands), June 6, 2024. ORBi

3. Regularity of positional numeration systems without a dominant root. [5] Combinatorics, Automata and Number Theory 2025, CIRM (Marseille, France), October 2, 2025. ORBi