Normal numbers with digit dependencies

Veronica Becher (Universidad de Buenos Aires)

Apr 22. 2025, 11:30 — 12:00

Given an integer 𝑏>0 and a set P of prime numbers, the set T_P of Toeplitz numbers comprises all elements of [0,𝑏) whose digits (π‘Ž_𝑛)_{n>0} in the base-𝑏 expansion satisfy π‘Ž_𝑛 = π‘Ž_𝑝𝑛 forall 𝑝 in P and 𝑛>0.   Using a completely additive arithmetical function, we construct a number in T_P that is simply Borel normal if, and only if the sum of 1/𝑝  over all 𝑝 in P diverges;  we provide an effective bound for the discrepancy.  For finite P we show that almost every number in T_P is Borel normal to base 𝑏.  For P={2} we show more: almost every number in T_P is Borel normal to all integer bases.  
Joint work  partly with  Christoph Aistleitner and  Olivier Carton, and partly with Agustín Marchionna and Gérald Tenenbaum.


 

Further Information
Venue:
ESI Boltzmann Lecture Hall
Files:
Slides
Associated Event:
Uniform Distribution of Sequences (Workshop)
Organizer(s):
Henk Bruin (U of Vienna)
Robbert Fokkink (TU Delft)
JΓΆrg Thuswaldner (MontanuniversitΓ€t Leoben)