Extremely non-normal numbers

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.