Transcendence of Sturmian or Morphic Continued Fractions

Abstract

We prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values, and begins with arbitrary long blocks which are ''almost squares,'' then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence (1+ (NnaM mod 2)) n \ 0 , or are a ''repetitive'' fixed point of a binary morphism satisfying some technical conditions.

Initial powers of Sturmian sequences

“…Let us prove the converse. Let α be as in the statement of the theorem, that is, for each pair of positive integers (s, t) with s > 1 there are only finitely many indices k for which (3) (a k , a k+1 ) = (s, t) or (a k , a k+1 , a k+2 ) = (1, 1, t).…”

“…There are a number of recent papers on powers of words occurring in Sturmian sequences (see for instance [1,2,3,6,8,9,16,17,27,32,38,41]). Quantities of interest include the supremum of powers of factors of a sequence (the index or critical exponent of the sequence), and the limit superior of powers of longer and longer factors of the sequence.…”

“…In [3] it is shown that each Sturmian sequence begins in infinitely many squares (see also [16]), and hence ice(ω) ≥ 2 for all Sturmian sequences ω. We show that the value 2 is attainable, and give the following characterization of those slopes for which there is a Sturmian sequence with initial critical exponent equal to 2: Theorem 1.1.…”

Continued Fractions With Bounded Partial Quotients

“…As much of the first part is easily derivable from the n = 2 case described in some detail in [1], we will be brief in our exposition.…”

“…If (u m ) m 1 is an infinite sequence with u m ∈ Σ for each m 1, then we let u be the infinite word u 1 and that this bound can be attained. If we assume that |Σ| = 2 and that u takes each of the two values with a frequency (see Definition 3.1), then the authors of [1] have shown that L(u) < 1.13, irrespective of the particular elements of Σ. In § § 2 and 3, we extend this result to any finite set, on the assumption that u takes each of the values in Σ with a frequency.…”