Balanced rectangles over Sturmian words

Wednesday, 21 October 2026 at 15:35

A Number Theory, Algebra and Geometry Seminar

Event details

The famous infinite Fibonacci word $01001010\cdots$ has many interesting properties. For example, any two blocks of the same length have either the same sum, or their sums are off by at most $1$. Words with this property are called balanced, and it turns out that the balanced words are precisely the Sturmian words.

Recently, Anselmo et al.\ considered ``rectangles'' of the Fibonacci word, where the Fibonacci word is shifted and aligned in an infinite matrix. They proved that for certain pairs $(m,n)$ the $m\times n$ rectangles are balanced as well. On the other hand, results by Berth\'e and Tijdeman show that this cannot be true for all $(m,n)$. In this talk, we fully characterise those $(m,n)$ for which the rectangles are balanced. Using continued fractions and Ostrowski representations, we can in fact do this for all Sturmian words.

Organiser

Mathematics and Statistics

Location

Harrison 170