On the irrationality measure of the Thue–Morse constant

Abstract

We provide a non-trivial measure of irrationality for a class of Mahler numbers defined with infinite products which cover the Thue-Morse constant. Among the other things, our results imply a generalization to [10].

On the spectrum of irrationality exponents of Mahler numbers

“…Therefore for s 3 − s = 0 we have that Φ(g a ) contains a primitive gap [1,3] of size 2. Note that z 3 m − s is always coprime with the polynomial z 2 + sz + s 2 .…”

“…are always coprime. Therefore the all values r c,m equal zero and equations (24) imply that for the gaps [u n , v n ] generated by [1,3] we have v n+1 u n+1 = 3v n − 2 3u n and therefore lim inf…”

“….} where the gap [3,7] is not primitive and is generated by [1,3]. Hence there are no other primitive gaps [u, v] with u < 7 and Theorem 2 implies:…”

“…We thus have that the numerator and the denominator of m−1 t=0 (z 2•3 t + sz 3 t + s 2 ) z 3 m − s are always coprime. Therefore the all values r c,m equal zero and equations (24) imply that for the gaps [u n , v n ] generated by [1,3] we have…”

Irrationality exponent and convergence exponent in continued fraction expansions

“…This improves the results in [14, theorem 4.1] where only an upper bound of the irrationality exponent was provided. Despite the irrationality exponents of Mahler numbers have been studied rather extensively, there is little to know about their more refined Diophantine approximation properties, we refer to the works of Badziahin and Zorin [3,5].…”