Boris Adamczewski: Furstenberg's conjecture, Mahler's method, and finite automata (NTWS 074)
Channel: Number Theory Web Seminar
Date: 2023-08-06

Abstract: It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.

Link to slides: https://drive.google.com/file/d/1Gm6f6ziHFF7GldABgp3b9FKYHsvz4cDK

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, April 1, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 2, 2021 (2am AEDT, 4am NZDT)