On the spectrum of irrationality exponents of Mahler numbers

Abstract

We consider Mahler functions f (z) which solve the functional equationB(z) ∈ Q(z) and d 2 is integer. We prove that for any integer b with |b| 2 either f (b) is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for f (b) as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao [6] where only an upper bound for the irrationality exponent was provided.

Irrationality exponent and convergence exponent in continued fraction expansions

“…In fact, v(x) ⩽ 2 holds for some transcendental numbers, and due to the theorem of Dirichlet, the upper bound is sharp. While for the case v(x) > 2, Badziahin [4] overcomed the obstacle and presented the precise value of v(x) for some Mahler numbers. This improves the results in [14, theorem 4.1] where only an upper bound of the irrationality exponent was provided.…”

Criteria for apwenian sequences

“…Let ξ be an irrational number. Its irrationality exponent is the supremum of real numbers ν such that ξ − r s < 1 s ν holds for infinitely many pairs of (r, s) ∈ Z × N. For the series f p (z) satisfying (1.1), Badziahin [4] showed that the irrationality exponent of f p (1/b) is rational, where b ≥ 2 is an integer with P (1/b p m ) = 0 for all integers m ≥ 0. To approach the exact value of the irrationality exponent, Badziahin's method requires to know the whole continued expansion of f p which is usually difficult.…”