On the rational approximation to the Thue–Morse–Mahler number

has infinitely many solutions in rational numbers p/q. It follows from the theory of continued fractions that μ(ξ) is always greater than or equal to 2, and an easy covering argument shows that μ(ξ) is equal to 2 for almost all real numbers ξ (with respect to the Lebesgue measure). Furthermore, Roth’s theorem asserts that the irrationality exponent of every algebraic irrational number is equal to 2. It is in general a very difficult problem to determine the irrationality exponent of a given transcendental real number ξ. Apart from some numbers involving the exponential function or the Bessel function (see the end of Section 1 of [1]) and apart from more or less ad hoc constructions (see below), there do not seem to be examples of transcendental numbers ξ whose irrationality exponent is known. When they can be applied, the current techniques allow us only to get an upper bound for μ(ξ). Clearly, the irrationality exponent of ξ can be read on its continued fraction expansion. But when ξ is defined by its expansion in some integer base b ≥ 2, we do not generally get enough information to determine the exact value of μ(ξ). More precisely, write