Rational digit systems and generalized Rauzy fractals

A numeration system is a way of writing numbers down as strings of digits in relation to a base. We typically use the decimal systems, but other bases such as 2 or 60 are also well known. Numeration systems can also exist in settings with more dimension: for example, complex numbers have two coordinates, and therefore they are depicted in a plane. Using a base that is a complex number allows us to express points of the plane as string of digits as well. Working with more dimensions has a nice outcome: we encounter that numeration systems are closely related to figures known as fractals. Fractals are intricate shapes with the property that, when zooming in on them, one can find similar patters occurring at infinitely small scales; they can be very beautiful as well. When working with a numeration system, one could associate it with a fractal set whose geometric properties shed light on the underlying arithmetic properties of the system. This project centers around a particular type of numeration system where the base is an algebraic number, that is, a complex number that satisfies a relatively simple equation. Some of the questions that we intend to answer are: how can we expand a complex number in an algebraic base? What do the digit expansions look like? When are these expansions unique, finite or periodic? How can we associate a fractal set to this kind of numeration system and what properties does the fractal satisfy? In order to answer these questions, we make use of a special type of numbers known as p-adic numbers. They are other kinds of digit expansions that are of interest among mathematicians, namely those obtained through continued fraction algorithms or through symbolic substitutions. These expansions are also related to fractals, in particular to a family known as Rauzy fractals. A central line of the project is to enlarge this family by considering a particular case of substitutions known as nonunimodular substitutions. Ideally, these substitutions and the numerations systems in algebraic bases that we mentioned before could be brought together under a common framework. A further goal of the project would be to come up with three dimensional illustrations of the fractals by using 3D printers and laser cutters. A nice feature of these sets is that they often fit together to form a puzzle, which can be fun to play with and a nice way to introduce people to the topic.