NON CONVERGING CONTINUED FRACTIONS RELATED TO THE STERN DIATOMIC SEQUENCE

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Boris Adamczewski

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NON-CONVERGING CONTINUED FRACTIONS RELATED TO THE STERN DIATOMIC SEQUENCE by Boris Adamczewski Abstract. — This note is essentially an addendum to the recent article of Dilcher and Stolarsky [7] though some results presented here may be of independent interest. We prove the transcen- dence of some irregular continued fractions which are related to the Stern diatomic sequence. The proofs of our results rest on the so-called Mahler method. 1. Introduction Given an integer a ≥ 2, it was recently observed in [7] that the regular continued fraction (1.1) C(a) = a+ 1 a2 + 1 a4 + 1 . . . + 1 a2n + 1 . . . , denoted in the sequel as usual by [a, a2, a4, . . . , a2 n , . . .], is transcendental. This is a con- sequence of Roth's theorem and follows directly from a result of Davenport and Roth [5] concerning the growth of denominators of convergents to an algebraic number. Quite surpris- ingly, the author of the present note was not able to pick up the scent of this simple example in the older literature though a function field analogue previously appeared in [12]. Indeed, viewed as a Laurent series in F2((1/x)), the continued fraction C(x) has the remarkable prop- erty of being a cubic element over the field F2(x).

stern diatomic

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transcendental over

following functional

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rational functions

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Functional equation Rational function

NON-CONVERGING CONTINUED FRACTIONS RELATED TO THE STERN DIATOMIC SEQUENCE

BorisAdamczewski

Abstract. —This note is essentially anaddendumto the recent article of Dilcher and [7We prove the] though some results presented here may be of independent interest. dence of some irregular continued fractions which are related to the Stern diatomic The proofs of our results rest on the so-called Mahler method.

Stolarsky transcen-sequence.

1. Introduction Given an integera≥2, it was recently observed in [7] that the regular continued fraction 1 (1.1)C(a) =a+, 1 2 a+ 1 4 a+ . 1 . . + 1 n 2 a+ . . . n 2 4 2 denoted in the sequel as usual by [, a a, a , . . ., . . . , a ], is transcendental. This is a con-sequence of Roth’s theorem and follows directly from a result of Davenport and Roth [5] concerning the growth of denominators of convergents to an algebraic number. Quite surpris-ingly, the author of the present note was not able to pick up the scent of this simple example in the older literature though a function ﬁeld analogue previously appeared in [12]. Indeed, viewed as a Laurent series inF2((1/x)), the continued fractionC(x) has the remarkable prop-erty of being a cubic element over the ﬁeldF2(xprecisely, it is the unique root in). More F2((1/x)) of the polynomial 3 2 t+xt+ 1. This follows from a simple computation using the fact that squaring here has a very trans-2 2 2 parent eﬀect: iff(x) = [a1(x), a2(x), . . .]∈F2((1/x)) thenf(x) = [a1(x), a2(x), . . .].

Key words and phrases. —Continued fractions, Stern diatomic sequence, Mahler’s method. 2000 Mathematics Subject Classiﬁcation. —11J70,11J81,11B39.

The author is supported by the ANR through the project “DyCoNum”–JCJC06 134288.

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