The project is based on three lines of research concerning fundamental number theoretical problems that have baffled mathematicians for decades. These include the study of integer base expansions of classical constants, arithmetical linear differential equations and their link with enumerative combinatorics, and arithmetic in positive characteristic. In this ERC project, we have obtained a wide variety of results concerning these three line of research.
We have made progress towards the Hartmanis-Stearns problem, in particular proving that the decimal expansion of an algebraic irrational numbers cannot be generated by a deterministic pushdown automaton. We have completely reworked Mahler's method, which is a method in transcendental number theory introduced by Mahler at the end of 1920s. In this direction, we have solved the main problems concerning the transcendence and algebraic independence of the values of Mahler functions at algebraic points. As a result, we proved that an irrational real number cannot be generated by a finite automata in two multiplicatively independent bases. This is part of a major conjecture by Furstenberg in the 1960s. This ERC project
has certainly contributed to the renewed interest for Mahler's equations.
We have developed a new method to proving the algebraic independence of G-functions, which form a class of fascinating analytic functions introduced by Siegel. They appear simultaneously in number theory, enumerative combinatorics, arithmetic geometry, and mathematical physics. Usually, the algebraic relations between such functions are studied through differential Galois theory. Instead, our new approach is based on the use of an
infinite family of linear difference equations associated with the Frobenius operator, which are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the corresponding G-functions satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients (also related to automatic sequences). Surprisingly, we shave hown that this situation occurs remarkably often.
We also used the notion of Strong Frobenius structure to study the notion of algebraicity modulo p, especially for generalized hypergeometric series and diagonals of algebraic power series.
We developed Mahler's method in positive characteristic, proving the analog of Nishioka's theorem in this framework, as well as its refinement
to linear relations. We have shown that a certain class of p-automatic sets plays a prominent role in the study of arithmetic geometry in characteristic p, especially with respect to various versions of the Mordell-Lang theorem. This approach based on automatic sets has the advantage of providing algorithms that allow solving effectively certain problems of Diophantine geometry in characteristic p.
In addition to the three line of research envisaged at the beginning of the project, several new directions have emerged in the course of this ERC project. The main example of this type is the development of Galois theories associated with linear difference equations and with linear differential equations to study the algebraic relations between generating series occurring in enumerative combinatorics or associated with automatic sequences.
Our results in this field have contributed to the emergence of a new research direction which is currently very active.
