Research Overview

I study hidden regularities in numbers and symbolic sequences from the viewpoints of number theory and dynamical systems. For example, I am interested in questions such as how uniformly a given sequence is distributed, and how the complexity of a symbolic sequence grows.

Such objects can often be defined in an elementary way. However, understanding their long-term behavior rigorously is frequently a difficult problem. One of the main goals of my research is to reveal the precise mathematical structures behind sequences and symbolic objects that may appear simple at first sight.

The uniform distribution of sequences is studied in a field called uniform distribution theory. This field investigates how various sequences are distributed. For instance, it is conjectured that in the decimal expansion of pi,

the digits from 0 to 9 appear with equal frequency; in other words, pi is conjectured to be a normal number. This remains an open problem.

Within uniform distribution theory, my research focuses especially on the following two topics:

• the behavior of digits appearing in beta-expansions of real numbers;
• the behavior of the fractional parts of geometric progressions whose common ratio is an algebraic number.