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→∞ x− 1 N (α; b, a, x),(2) if the limit exists. 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 integer a∈[0, b). When α is a b-adic fraction a/bn, which has one b-ary expansion with all but finitely many digits equaling zero and another b-ary expansion with all but finitely many digits equaling b− 1, these limiting frequencies are not uniquely defined; however, such an α is not simply normal to the base b in either case.

A number is normal to the base b if it is simply normal to each of the bases b, b2, b3,.... This is equivalent to demanding that for any finite string a1a2... ak of base-b digits, the limiting frequency of occurrences of this string in the b-ary expansion of α (analogous to (2)) exists and equals 1/bk; see [6, Chapter 8]. Champernowne showed [1] that the number 0. 12345678910111213... formed by concatenating all of the positive integers together into a single decimal is normal to base 10 (an analogous construction works for any base b≥ 2), and this sort of example has been generalized; see [2] or [3]. It is known that almost all real numbers are normal to any given base b [6, Theorem 8.11], and consequently almost all real numbers are absolutely normal, ie, normal to all bases b≥ 2 simultaneously. On the other hand, no one has proved a single naturally occurring real number to be absolutely normal. Let us call a number abnormal to the base b if it is not normal to the base b, and absolutely abnormal if it is abnormal to all bases b≥ 2 simultaneously. For example, every rational number r is absolutely abnormal: any b-ary expansion of r eventually repeats, say with period k, in which case r is about as far from being simply normal to the base bk as it can be. Even though the set of absolutely abnormal numbers is the intersection of countably many sets of measure zero, Maxfield has pointed out that the set of absolutely abnormal numbers is uncountable and dense in the real line [5]; later, Schmidt gave a complicated constructive proof of this fact [7]. In this paper we exhibit a simple construction of a specific irrational (in fact, transcendental) real number that is absolutely abnormal. In fact, our construction easily generalizes to a construction giving uncountably many absolutely abnormal numbers in any open interval. It is instructive to consider why constructing an irrational, absolutely abnormal number is even difficult. Since we already know that rational numbers are absolutely abnormal, our first thought might be to choose an irrational number whose b-ary expansions mimic those of rational numbers for long stretches, ie, an irrational number