Constructions of normal numbers with infinite digit sets

Journal of Complexity, vol. 89, pp. 101945

Abstract

Let L = ( L d ) d ∈ N be any ordered probability sequence, i.e., satisfying 0 < L d + 1 ≤ L d for each d ∈ N and ∑ d ∈ N L d = 1 . We construct sequences A = ( a i ) i ∈ N on the countably infinite alphabet N in which each possible block of digits α 1 , … , α k ∈ N , k ∈ N , occurs with frequency ∏ d = 1 k L α d . In other words, we construct L -normal sequences. These sequences can then be projected to normal numbers in various affine number systems, such as real numbers x ∈ [ 0 , 1 ] that are normal in GLS number systems that correspond to the sequence L or higher dimensional variants. In particular, this construction provides a family of numbers that have a normal Lüroth expansion.

Authors 2

1. Aafko Boonstra corresponding

   Vrije Universiteit Amsterdam

   Affiliation as printed

   Department of Mathematics, VU University Amsterdam, De Boelelaan 1111, 1081HV Amsterdam, the Netherlands

2. Charlene Kalle Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands