Construction of normal numbers with respect to Generalized L\"uroth Series from equidistributed sequences

TL;DR

This paper presents a novel method for constructing normal numbers relative to Generalized Lüroth Series by concatenating segments of equidistributed sequences, extending the class of known normal numbers.

Contribution

It introduces a new construction technique for normal numbers with respect to Generalized Lüroth Series, including those with infinite digit sets.

Findings

Constructed normal numbers using equidistributed sequences

Applicable to Generalized Lüroth Series with infinite digit sets

Method extends previous constructions of normal numbers

Abstract

Generalized L\"uroth series generalize $b$-adic representations as well as L\"uroth series. Almost all real numbers are normal, but it is not easy to construct one. In this paper, a new construction of normal numbers with respect to Generalized L\"uroth Series (including those with an infinite digit set) is given. Our method concatenates the beginnings of the expansions of an arbitrary equidistributed sequence.