Liouville Number - Irrationality Measure

The irrationality measure (or irrationality exponent or approximation exponent or Liouville–Roth constant) of a real number x is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any n in the power of q, we find the least upper bound of the set of real numbers μ such that

is satisfied by an infinite number of integer pairs (p, q) with q > 0. This least upper bound is defined to be the irrationality measure of x. For any value μ less than this upper bound, the infinite set of all rationals p/q satisfying the above inequality yield an approximation of x. Conversely, if μ is greater than the upper bound, then there are at most finitely many (p, q) with q > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of q. In other words, given the irrationality measure μ of a real number x, whenever a rational approximation x ≅ p/q, p,q ∈ N yields n + 1 exact decimal digits, we have

except for at most a finite number of “lucky” pairs (p, q).

For a rational number α the irrationality measure is μ(α) = 1. The Thue–Siegel–Roth theorem states that if α is an algebraic number, real but not rational, then μ(α) = 2.

Almost all numbers have an irrationality measure equal to 2.

Transcendental numbers have irrationality measure 2 or greater. For example, the transcendental number e has μ(e) = 2. The irrationality measure of π is at most 7·60630853: μ(log 2)<3·57455391 and μ(log 3)<5·125.

The Liouville numbers are precisely those numbers having infinite irrationality measure.