Irrationality Measures of log 2 and π/√3

1. Irrationality Measures An irrationality measure of x ∈ R \Q is a number μ such that ∀ > 0,∃C > 0,∀(p, q) ∈ Z, ∣∣∣∣x− pq ∣∣∣∣ ≥ C qμ+ . This is a way to measure how well the number x can be approximated by rational numbers. The measure is effective when C( ) is known. We denote inf {μ | μ is an irrationality measure of x } by μ(x), and we call it the irrationality measure of x. By definition, rational numbers do not have an irrationality measure. Given two irrationality measures for a number, the smaller one is more precise, since it shows the number to be further “away” from rational numbers. For all x ∈ R \ Q, the inequality μ(x) ≥ 2 holds and gives the minimal possible value. This inequality follows from a pigeon-hole principle: for any integer n > 1, the fractional parts {qx}, 0 ≤ q < n together with the number 1, are n + 1 real numbers in the interval [ 0, 1 ]; therefore two of them must be at distance less than or equal to 1/n; their difference is of the form qx − p, so that |x − p/q| < 1/nq < 1/q2. A more explicit construction of these rational approximations is given by continued fraction expansions. The periodicity of continued fraction expansions of irrational quadratic numbers implies that they have an (effective) measure equal to 2. This result was generalized by Liouville in 1844, when he obtained the first practical criterion for constructing transcendental numbers. Theorem 1 (Liouville). An algebraic number α of degree n has effective irrationality measure n. Proof. Let P be the minimal polynomial of α. This is a polynomial of degree n with integer coefficients. By the mean value theorem, P (α)− P (p/q) = −P (p/q) = (α− p/q)P ′(ξ), for some ξ between α and p/q. Since P is irreducible, P (p/q) 6= 0 and ∣∣qnP (p/q)∣∣ is an integer which is therefore at least 1. It is sufficient to restrict attention to p/q at distance less than 1 from α. Then P ′(ξ) has a lower bound on this interval and this proves the measure. The bound is made effective in terms of the height H of P (the largest absolute value of its coefficients), as ∣∣P ′(ξ)∣∣ < n2H(1 + |α|)n−1. 102 Irrationality Measures of log 2 and π/ √ 3 Number log 2 π π/ √ 3 ζ(2) ζ(3) measure 3.8913997 8.016045 4.601579 5.441243 5.513891 author Rukhadze (1987) Hata (1993) Rhin & Viola (1996) Table 1. Irrationality measures and their authors. Using this result, Liouville constructed so-called Liouville numbers whose smallest measure is infinite. These numbers are therefore transcendental, since their measure cannot be bounded by any integer as demanded by the above theorem. A family of such numbers is ∑ n≥0 a−n!, a ∈ N \ {0, 1}. Indeed, truncating after the kth term gives a rational approximation pk/qk with qk = ak! and a simple computation on the tail of the series shows that it is less than q−k k . In the twentieth century, a sequence of results improved on Liouville’s theorem, this was ended by Roth, who showed in 1955 that all algebraic numbers have irrationality measure exactly 2 (this result is not effective). In a different direction, Khintchine showed that almost all real numbers (in the sense of Lebesgue) have irrationality measure 2. However, not all reals have measure 2: apart from Liouville numbers, for every μ ∈ [ 2,∞) the following gives a family of numbers with measure exactly μ: [a] + 1