Outstanding Challenges in Combinatorics on Words Feb

S (in alphabetic order by speaker surname) Speaker: Boris Adamczewski (Institu Camille Jordan & CNRS) Title: Combinatorics on words and Diophantine approximation Abstract: A very fruitful interplay between combinatorics on words and Diophantine approximation comes up with the use of numeration systems. Finite and infinite words occur naturally in Number Theory when one considers the expansion of a real number in an integer base or its continued fraction expansion. Conversely, with an infinite word w on the finite alphabet {0, 1, . . . , b − 1} one can associate the real number ξw whose base-b expansion is given by w. Many problems are then concerned with the following question: how the combinatorial properties of the word w and the Diophantine properties of the number ξw may be related? In this talk, I will survey some of the recent advances on this topic. I will also try to point out new challenges. Speaker: Golnaz Badkobeh (King’s College London) Title: Fewest repetitions vs maximal-exponent powers in infinite binary words Abstract: The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at most r(a). This notion was introduced in 1972 by Dejean who gave the exact values of r(a) for every alphabet size a as it has been eventually proved in 2009. The finite-repetition threshold for an a-letter alphabet refines the above notion. It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contain a finite number of factors with exponent r(a). It is known from Shallit (2008) that FRt(2) = 7/3. With each finite-repetition threshold is associated the smallest number of r(a)-exponent factors that can be found in the corresponding infinite word. It has been proved by Badkobeh and Crochemore (2010) that this number is 12 for infinite binary words whose maximal exponent is 7/3. In this article we give some new results on the trade-off between the number of squares and the number of maximal-exponent powers in infinite binary words, in the three cases where the maximal exponent is 7/3, 5/2, and 3. These are the only threshold values related to the question. Speaker: Nicolas Bédaride (Aix-Marseille University) Title: Piecewise isometries and words Abstract: A piecewise isometrie of the plane is a bijective map defined on the complement of several lines such that the restriction to a connected set is an isometry of the plane. In this talk, I will describe some classical piecewise isometries of the plane, and the associated language obtained by a coding. Speaker: Jason Bell (Simon Fraser University) Title: On Automatic Sequences Speaker: Srecko Brlek (UQAM Laboratoire de Combinatoire et d’informatique Mathématique) Title: The last talk on the Kolakoski sequence