The Irrationality Exponents of Computable Numbers

The irrationality exponent a of a real number x is the supremum of the set of real numbers z for which the inequality 0 < |x − p/q | < 1/qz is satisfied by an infinite number of integer pairs (p, q) with q> 0. Rational numbers have irrationality exponent equal to 1, irrational numbers have it greater than or equal to 2. The Thue–Siegel–Roth theorem states that the irrationality exponent of every irrational algebraic number is equal to 2. Almost all real numbers (with respect to the Lebesgue measure) have irrationality exponent equal to 2. The Liouville numbers are precisely those numbers having infinite irrationality exponent. For any real number a greater than or equal to 2, Jarník (1931) used the theory of continued fractions to give an explicit construction, relative to a, of a real number xa such that the irrationality exponent of xa is equal to a. For a = 2, we can take x2 = 2. For a> 2, we construct inductively the sequence of partial quotients of xa = [0; a1, a2,...]. For n ≥ 1, set pn/qn = [0; a1, a2,..., an]. Take a1 = 2 and an+1 = bqa−2n c, for n ≥ 1, where b·c denotes th