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

Abstract

Let p be a prime number and ξ an irrational p-adic number. Its multiplicative irrationality exponent µ × (ξ) is the supremum of the real numbers µ × for which the inequalityhas infinitely many solutions in nonzero integers a, b. We show that µ × (ξ) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of ξ. We establish that µ × (ξ t,p ) = 3, where ξ t,p is the p-adic number 1 − p − p 2 + p 3 − p 4 + . . ., whose sequence of digits is given by the Thue-Morse sequence over {−1, 1}.

Morphic and automatic words: maximal blocks and Diophantine approximation

“…Furthermore, it is proved in [11] that, for b ≥ 2, the irrationality exponent of the real number whose sequence of base-b digits is the Thue-Morse sequence on {0, 1} is equal to 2.…”

Algebraic independence of the generating functions of Stern’s sequence and of its twist

“…has only finitely many solutions (p, q) ∈ Z × N. For certain rational α's, we can get a more precise and rather sharp result by using the ideas of the works [1] and [5]. Namely the following result holds.…”

“…First we note that the proof of Theorem 1.5 follows immediately from the following special case of [14 To prove Theorem 1.6, we use Padé approximations of A and B similarly to the procedure in [1] and [5] for the generating function of the Thue-Morse sequence. In our case, we do not have an analogous non-vanishing result but we compute several low degree Padé approximants and iterate these by the functional equation of A or B.…”

Hankel continued fraction and its applications

“…In 1998, Allouche, Peyrière, Wen and Wen proved that all Hankel determinants of the Thue-Morse sequence are nonzero [APWW]. Bugeaud [Bu11] was able to prove that the irrationality exponent of the Thue-Morse-Mahler number is equal to 2 by using APWW's result. Using Bugeaud's method, several authors obtained the following results: first, Coons [Co13] who proved that the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2; then, Guo, Wu and Wen who showed that the irrationality exponents of the regular paperfolding numbers are exactly 2 [GWW].…”