# Extremely non-normal numbers

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## Abstract

We prove that from a topological point of view most numbers fail to be normal in a spectacular way. For an integer N greater than or equal to 2 and x is an element of [0, 1], let x = Sigma(n=1)(infinity) epsilon(N,n)(x)/N-n, where epsilon(N,n)(x) is an element of {0, 1, ..., N-1} for all n, denote the unique non-terminating N-adic expansion of x. For a positive integer n and a finite string i = i(1) (. . .) i(k) with entries i(j) is an element of {0, 1, ..., N-1}, we write

Pi(N)(x, i; n) = \{1 less than or equal to i less than or equal to n\epsilon(N,i)(x) = i(1), ..., epsilon(N,i+k-1)(x) = i(k)}\/n

for the frequency of the string i among the first n digits in the N-adic expansion of x, and let Pi(N)(k) (x; n) = (Pi(N)(x, i; n))(i) denote the vector of frequencies Pi(N)(x, i; n) of all strings i = i1 (. . .) i(k) of length k with entries i(j) is an element of {0, 1, ..., N-1}. We say that a number x is extremely non-normal if each shift invariant probability vector in R-Nk is an accumulation point of the sequence (Pi(N)(k) (x; n))(n) simultaneously for all k and all bases N, and we denote the set of extremely non-normal numbers by E, i.,e.

E = [GRAPHIC][GRAPHIC]{x is an element of [0, 1]\ each p is an element of Gamma(N)(k)

is an accumulation point of the sequence (Pi(N)(x, i; n))(n)},

where GammaNk denotes the simplex of shift invariant probability vectors in R-Nk. Our main result says that E is a residual set, i.e. the complement [0,1]\E is of the first category. Hence, from a topological point of view, a typical number in [0, 1] is as far away from being normal as possible. This result significantly strengthens results by Maxfield and Schmidt. We also determine the Hausdorff dimension and the packing dimension of E.

Original language English 43-53 11 Mathematical Proceedings of the Cambridge Philosophical Society 137 1 Published - Jul 2004

## Keywords

• DIVERGENCE POINTS
• SETS

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