Mahler's method in several variables and finite automata

Abstract

We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent matrix transformations. Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018. The main new feature is that they apply now without any restriction on the matrices defining the corresponding Mahler systems. As a consequence, we settle several problems concerning expansions of numbers in multiplicatively independent bases. For instance, we prove that no irrational real number can be automatic in two multiplicatively independent integer bases, and we give a new proof and a broad algebraic generalization of Cobham's theorem in automata theory. We also provide a new proof and a multivariate generalization of Nishioka's theorem, a landmark result in Mahler's method. Contents1. Introduction 2. Mahler's method in several variables 3. Notation 4. Admissibility conditions 5. A new vanishing theorem 6. Mahler's method in families 7. Hilbert's Nullstellensatz and relation matrices 8. Proof of Theorem 6.2 9. Proofs of Theorems 2.3, 2.6, 2.8, and of Corollaries 2.5 and 2.9 10. Proof of Theorem 1.1 Appendix A. Representing numbers in independent bases References

Slices and distances: on two problems of Furstenberg and Falconer

“…Unfortunately, this says nothing about points x for which dim H (O a,x ) = 0, since all such points form a zero dimensional set. Recently, B. Adamczewski and C. Faverjon [1] showed that an irrational number cannot be automatic in bases a and b; being automatic is a computational notion of "simplicity", and so this can be seen as a first verification that an irrational number cannot be "too simple" in two multiplicatively independent bases.…”