Words and Transcendence

Is it possible to distinguish algebraic from transcendental real numbers by considering the b-ary expansion in some base b 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g 2, the g-ary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g 3 is an integer, a a digit in {0, . . . , g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the g-ary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the g-ary expansion of real numbers, where g 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g 2 be an integer. Any real number x has a unique expansion x = a−kg + · · ·+ a−1g + a0 + a1g−1 + a2g−2 + · · · , where k 0 is an integer and the ai for i −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1, . . . , g− 1}. Uniqueness is subject to the condition that the sequence (ai)i −k is not ultimately constant and equal to g − 1. We write this expansion x = a−k · · · a−1a0.a1a2 · · · . Example. We have √ 2 = 1.41421356237309504880168872420 . . . in base 10 (decimal expansion), whereas √ 2 = 1.01101010000010011110011001100111111100111011110011 . . . in base 2 (binary expansion). The first question in this direction is whether each digit always occurs at least once. 450 MICHEL WALDSCHMIDT Conjecture 1.1. Let x be an real irrational algebraic number, g 3 a positive integer and a an integer in the range 0 a g− 1. Then the digit a occurs at least once in the g-ary expansion of x. For g = 2 it is obvious that each of the two digits 0 and 1 occurs infinitely often in the binary expansion of an irrational algebraic number. The same is true for each of the sequences of two digits 01 and 10. Conjecture 1.1 implies that each of the sequences of digits 00 and 11 should also occur infinitely often in such an expansion; apply Conjecture 1.1 with g = 4. There is no explicitly known example of a triple (g, a, x), where g 3 is an integer, a a digit in {0, . . . , g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the g-ary expansion of x. Another open problem is to produce an explicit pair (x, g), where g 3 is an integer and x a real irrational algebraic number, for which we can claim that the number of digits which occur infinitely many times in the g-ary expansion of x is at least 3. Even though few results are known and explicit examples are lacking, something is known. For any g 2 and any k 1 there exist real algebraic numbers x such that any sequence of k digits occurs infinitely often in the g-ary expansion of x. However the connection with algebraicity is weak: indeed Mahler proved in 1973 more precisely that for any real irrational number α and any sequence of k digits in the set {0, . . . , g − 1}, there exists an integer m for which the g-ary expansion of mα contains infinitely many times the given sequence; see [3, Theorem M] and [50]. According to Mahler, the smallest such m is bounded by g2k+1. This estimate has been improved by Berend and Boshernitzan [26] to 2gk+1 and one cannot get better than g − 1. If a real number x satisfies Conjecture 1.1 for all g and a, then it follows that for any g, each given sequence of digits occurs infinitely often in the g-ary expansion of x. This is easy to see by considering powers of g. Borel asked more precise questions on the frequency of occurrences of sequences of binary digits of real irrational algebraic numbers. We need to introduce some definitions. First, a real number x is called simply normal in base g if each digit occurs with frequency 1/g in its g-ary expansion. A very simple example in base 10 is x = 0.123456789012345678901234567890 . . . , where the sequence 1234567890 is repeated periodically, but this number is rational. We have x = 1234 567 890 9 999 999 999 = 137 174 210 1 111 111 111 . Second, a real number x is called normal in base g or g-normal if it is simply normal in base g for all m 1. Hence a real number x is normal in base g if and only if, for all m 1, each sequence of m digits occurs with frequency 1/g in its g-ary expansion. Finally, a number is called normal if it is normal in all bases g 2. Borel suggested in 1950 that each real irrational algebraic number should be normal. Conjecture 1.2 (Borel, 1950). Let x be a real irrational algebraic number and g 2 a positive integer. Then x is normal in base g. As shown by Borel [28], almost all numbers, in the sense of Lebesgue measure, are normal. Examples of computable normal numbers have been constructed by Sierpinski, Lebesgue, Becher and Figueira; see [24]. However, the known algorithms to compute such examples are fairly complicated; indeed, “ridiculously exponential”, according to [24]. WORDS AND TRANSCENDENCE 451 An example of a 2-normal number is the binary Champernowne number, obtained by concatenation of the sequence of integers 0. 1 10 11 100 101 110 111 100