@TECHREPORT{Ugalde98analternative, author = {Edgardo Ugalde}, title = {An Alternative Construction of Normal Numbers}, institution = {}, year = {1998} }

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Abstract

A new class of b{adic normal numbers is built recursively by using Eulerian paths in a sequence of de{Bruijn digraphs. In this recursion, a path is constructed as an extension of the previous one, in such way that the b{adic block determined by the path contains the maximum number of dierent b{adic subblocks of consecutive lengths in the most compact arrangement. Any source of redundancy is avoided at every step. Our recursive construction is an alternative to the several well{known concatenative constructions a la Champernowne. Keywords: Normal numbers, Eulerian paths. 1 Introduction Let b be a xed integer. A number x 2 [0; 1] is a b{adic normal if each block q(1)q(2) q(k) on b symbols appears in the b{adic expansion of x with frequency b n . In [Bo] Borel proved that the set of all b{adic normals is a set of Lebesgue measure equals to 1, but it was only 35 years after Borel's proof that Champernowne [Ch] presented the rst explicit normal number in base 10. He proceede...