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Equidistribution, fractal measures and arithmetic

Periodic Reporting for period 2 - EFMA (Equidistribution, fractal measures and arithmetic)

Reporting period: 2020-04-01 to 2021-09-30

The project has three main strands, which are the speed of mixing of random walks, self-similar fractals and arithmetic properties of random polynomials. These seemingly unrelated problems are, in fact, intimately connected by common underlying mathematical phenomena.

Generating a random object with a specified distribution is an important problem in mathematics with manifold applications in the sciences. One commonly used approach to this problem is a random walk, which starts with a non-random starting object, which is modified step by step in a random manner gradually increasing its randomness. Perhaps the most familiar instance of this is card shuffling. The order of the cards in the deck is modified in a random manner, making it more and more random after each shuffle. After sufficiently many steps of shuffling, the deck becomes truly random, each possible ordering of the cards becomes (almost) equally likely. It is an important question to understand how many shuffle does it take to make the deck random, and more generally how many steps of a random walk need to be taken to reach the desired target distribution. The mathematical notion, which makes this question precise is called the mixing time. One aim of the project is to understand the mixing times of certain types of random walks, which are endowed with an underlying algebraic structure. The interest in this problem is twofold. First understanding it will shed light on the underlying algebraic structure and on related problems in pure mathematics. Second, these random walks serve as "toy models" for more complex problems.

Self-similar fractals are arguably among the most natural families of fractals. They are characterised by the property that they contain exact scaled copies of themselves at all scales. The dimension of a fractal is a mathematical notion designed to measure its "size". It is one of the most basic problems in fractal geometry to determine the dimension of fractals, yet this is still far from properly understood even for self-similar fractals. One of the aims of the project is to address this problem.

The third and last strand of the project aims to understand number theoretic properties of random polynomials. One such property is irreducibility, that is whether or not a polynomial can be written as the product of two or more polynomials. This is analogous to the notion of prime numbers. Such questions are very interesting in their own right, but the primary reason for including them in the project is their intimate relation to the other two strands.
Perhaps the simplest random walk one may consider is the one on a circle moving by 1 unit in a randomly chosen direction in each step. This random walk has a very long mixing time, simply because it takes very long to move around the circle. Chung Diaconis and Graham suggested a way to reduce the mixing time by applying a simple operation to the position of the walker between the steps. They managed to show that this reduces the mixing time quite dramatically, but they were not able to determine the full extend of this improvement. The PI worked on this problem with Sean Eberhard and Emmanuel Breuillard in two different settings. Ongoing work of Oren Becker, Emmanuel Breuillard, Amitay Kamber and the PI aims to extend the new techniques introduced for this problem to random walks with more complex underlying algebraic structures.

The PI began the study of the dimension theory of self-similar fractals by the simplest example called Bernoulli convolutions. These are self-similar fractals, that can be defined by iterating two scaling maps. Ariel Rapaport and the PI worked on extending these techniques to more general fractals.
In the above mentioned works about the Chung Diaconis Graham random walk, we determined the precise mixing time of the random walk for typical values of the parameters in two different settings solving a long standing open problem. We hope that these techniques will shed new light on random walks with more complicated underlying structure such as semisimple linear groups.

The PI determined the dimension of Bernoulli convolutions for transcendental values of the parameters. Since the algebraic case was already quite well understood, this means that our understanding of the dimension theory of Bernoulli convolutions is now almost complete. This marks an important milestone in the theory. We managed to extend some of the theory to self-similar fractals defined by three scaling maps, and we hope that this can be extended further to cover more general fractals.