Periodic Reporting for period 4 - ANT (Automata in Number Theory) Reporting period: 2020-04-01 to 2022-03-31 Summary of the context and overall objectives of the project Finite automata are fundamental objects in Computer Science, of great importance on one hand for theoretical aspects (formal language theory, decidability, complexity) and on the other for practical applications (parsing). In Mathematics, and especially in number theory, finite automata are mainly used as simple devices for generating sequences of symbols over a finite set (e.g. digital representations of real numbers), and for recognizing some sets of integers or more generally of finitely generated abelian groups or monoids. One of the main features of these automatic structures comes from the fact that they are highly ordered without necessarily being trivial (i.e. periodic). With their rich fractal nature, they lie somewhere between order and chaos, even if, in most respects, their rigidity prevails. Over the last few years, several ground-breaking results have lead to a great renewed interest in the study of automatic structures in arithmetics. A primary objective of the ANT project is to exploit this opportunity by developing new directions and interactions between automata and number theory. The project is based on three lines of research concerning fundamental number theoretical problems that have baffled mathematicians for decades. They include the study of integer base expansions of classical constants, of arithmetical linear differential equations and their link with enumerative combinatorics, and of arithmetics in positive characteristic. At first glance, these topics may seem unrelated, but, surprisingly enough, the theory of finite automata serves as a natural guideline. We stress that this new point of view on classical questions is a key part of our methodology: we aim at creating a powerful synergy between the different approaches we propose to develop, placing automata theory and related methods at the heart of the subject. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far 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. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The main outcomes of this ERC projects concern old and notoriously difficult problems about the expansion of real numbers in integer bases. In particular, we have further developed a method in transcendental number theory introduced by Mahler at the end of 1920s, including its extension to several variables and several Mahler operators. As a result, we proved among other results 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. We have also developed a completely new approach to proving the algebraic independence of G-functions, which form a class of fascinating analytic functions introduced by Siegel. The latter uses reduction modulo p techniques instead of differential Galois theory, which is the classical approach in this framework. Finally, we have developed the use of certain Galois theories associated with linear difference equations and with linear differential equations to study problems in enumerative combinatoric and automata theory. This has opened a new research direction which is currently very active. hafermancarpet.jpg
