Abnormal Numbers

A normal number is one whose decimal expansion (or expansion to some base other than 10) contains all possible finite configurations of digits roughly with their expected frequencies. More formally, let N(α; b, a, x) = #{1 ≤ n ≤ x : the nth digit in the base-b expansion of α is a} (1) denote the counting function of the occurrences of the digit a (0 ≤ a < b) in the b-ary expansion of the real number α, and define the corresponding limiting frequency δ(α; b, a) = lim x→∞ 1 x N(α; b, a, x). (2) The number α is simply normal to the base b if the limit defining δ(α; b, a) exists and equals 1/b for each 0 ≤ a < b. A number is normal to the base b if it is simply normal to each of the bases b, b 2 , b 3 ,. .. . For instance, it was shown by Champernowne [1] that the number 0.12345678910111213. .. , formed by concatenating all of the positive integers together into a single decimal, is normal to base 10 (the analogous construction works for any base b ≥ 2), and this sort of example has been generalized [2, 3]. It is known that almost all real numbers are normal to any given base b (see for instance [6, Theorem 8.11]), and consequently almost all real numbers are absolutely normal , i.e., normal to all bases b ≥ 2 simultaneously. On the other hand, we do not know a single specific real number that is absolutely normal. Let us call a number abnormal to the base b if it is not normal to base b, and absolutely abnormal if it is abnormal to all bases b ≥ 2 simultaneously. For instance, any rational number is absolutely abnormal: the b-ary expansion of r will eventually repeat, say with period k, in which case r is about as far from being simply normal to the base b k as it can be. Even though the set of absolutely abnormal numbers is the intersection of countably many sets of measure zero, it was pointed out by Maxfield [5] that the set of numbers normal to a given base b is of first category in the sense of Baire, and therefore the absolutely abnormal numbers are uncountable and dense; however, this proof is not constructive. It is the purpose …