New York Journal of Mathematics

New York J. Math. 4 (1998) 31–34.

Irrational Numbers of Constant Type — A New Characterization

Manash Mukherjee and Gunther Karner

Abstract. Given an irrational number α and a positive integer m, the distinct fractional parts of α, 2α, ⋯, mα determine a partition of the interval [0, 1]. Defining d_{α}(m) and d′_{α}(m) to be the maximum and minimum lengths, respectively, of the subintervals of the partition corresponding to the integer m, it is shown that the sequence ( \frac{d_{α}(m)}{d′_{α}(m)} )_{m=1}^{∞} is bounded if and only if α is of constant type. (The proof of this assertion is based on the continued fraction expansion of irrational numbers.)

Contents

1. Introduction 31 2. Basic Properties of Continued Fractions 32 3. The Main Theorem 32 References 34

1. Introduction

Let α be a real irrational number, and α − [α] = {α} be the fractional part of α (where [·] is the greatest integer function). For k = 1, 2, ⋯, m, consider the sequence of distinct points {kα} in [0, 1], arranged in increasing order:

0 < {k_{1}α} < ⋯ < {k_{j}α} < {k_{j+1}α} < ⋯ < {k_{m}α} < 1

where 1 ≤ k_{j} ≤ m for j = 1, 2, ⋯, m.

Let d_{α}(m) and d′_{α}(m) denote, respectively, the maximum and minimum lengths of the subintervals determined by the above partition of [0, 1]. Using the continued fraction expansion of α (see Section 2), and the Three Distance Theorem (Theorem 1, Section 3), we obtain a new characterization of irrational numbers of constant type (defined as irrationals with bounded partial quotients). We show in

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

Received July 25, 1997.

Mathematics Subject Classification. 11A55.

Key words and phrases. Irrational numbers, Continued fractions.

Theorem 2 (The Main Theorem, Section 3), that the sequence (\frac{d_{α}(m)}{d′_{α}(m)})_{m=1}^{∞} is

bounded if and only if α is an irrational number of constant type.

Other characterizations of irrational numbers of constant type can be found in the survey article by J. Shallit [3]. In the investigation of certain dynamical systems, Theorem 2 is essential for the formulation of stability criteria for orbits of so-called quantum twist maps [2].

2. Basic Properties of Continued Fractions

Throughout this paper, ℕ, ℤ, ℚ, ℝ denote the natural numbers, integers, rational numbers, and real numbers, respectively, and α denotes an irrational number. Proofs of the facts 1 and 2 below can be found in [1, p. 30].

Fact 1. α ∈ ℝ\ℚ if and only if α has \underline{infinite} (simple) continued fraction expansion:

α = [a_{0}; a_{1}, a_{2}, ⋯, a_{n}, ⋯] = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + ...}}

where a_{0} ∈ ℤ and a_{n} ∈ ℕ for n ≥ 1. □

Definition 1. An irrational number, α, is of constant type provided there exists a positive number, B(α), such that B(α) = sup_{n≥1} (a_{n}) < ∞. (See reference [3].)

Fact 2. Define integers p_{n} and q_{n} by:

\begin{matrix}p_{−1} = 1 & ; & p_{0} = a_{0} & ; & p_{n} = a_{n}p_{n−1} + p_{n−2} & , n ≥ 1 \\ q_{−1} = 0 & ; & q_{0} = 1 & ; & q_{n} = a_{n}q_{n−1} + q_{n−2} & , n ≥ 1 \end{matrix}

Then, for n ≥ 0, gcd(p_{n}, q_{n}) = 1, and 0 < q_{1} < q_{2} < ⋯ < q_{n} < q_{n+1} < · · ·. Furthermore, (q_{n}α − p_{n}) and (q_{n+1}α − p_{n+1}) are of opposite sign for all n ≥ 0. □

Note: (\frac{p_{n}}{q_{n}} )_{n≥0} are called the principal convergents to α.

Lemma 1. Define η_{n} = |q_{n}α − p_{n}|. For all n ≥ 0, η_{n−1} = a_{n+1}η_{n} + η_{n+1}, and hence, η_{n} < η_{n−1}.

Proof. From Fact 2, we have

|q_{n−1}α − p_{n−1}| = |(q_{n+1}α − p_{n+1}) − a_{n+1}(q_{n}α − p_{n})|

The lemma follows from the fact that a_{n} > 0 for n ≥ 1, and that (q_{n}α − p_{n}) and (q_{n+1}α − p_{n+1}) have opposite signs. □

3. The Main Theorem

For α ∈ ℝ \ ℚ and m ∈ ℕ, the fractional parts, {α}, {2α}, …, {mα}, define a partition, P_{α}(m), of [0, 1]:

0 = d_{0} < d_{1} < ⋯ < d_{j} < d_{j+1} < ⋯ < d_{m} < d_{m+1} = 1

The maximum and minimum lengths of the subintervals of P_{α}(m) are denoted, respectively, by

d_{α}(m) := max_{0≤i≤m} (d_{i+1} − d_{i})
d′_{α}(m) := min_{0≤i≤m} (d_{i+1} − d_{i})

For the partition P_{α}(m), the differences (d_{i+1} −d_{i}) can be completely characterized [4] in terms of η_{n} = |q_{n}α − p_{n}|. Collecting the relevant results in reference [4], we have

Theorem 1 (Three Distance Theorem). Let α ∈ ℝ \ ℚ and m ∈ ℕ.

(a) m can be uniquely represented as m = rq_{k} + q_{k−1} + s, for some k ≥ 0, 1 ≤ r ≤ a_{k+1}, and 0 ≤ s < q_{k} (where a_{k}’s are the partial quotients of α and q_{k}’s are given in Fact 2).

(b) For the partition P_{α}(m), there are (r − 1)q_{k} + q_{k−1} + s + 1 subintervals of length η_{k}, s+1 subintervals of length η_{k−1} −rη_{k}, and q_{k} −(s+1) subintervals of length η_{k−1} − (r − 1)η_{k}, where the unique integers k, r and s are as in part (a).

Remark 1. From Theorem 1, we observe

(a) η_{k−1} − rη_{k} = η_{k+1} + (a_{k+1} − r)η_{k}, by Lemma 1

(b) η_{k−1} − (r − 1)η_{k} = η_{k} + η_{k−1} − rη_{k}

(c) When q_{k} = s + 1, there are no subintervals of length η_{k−1} − (r − 1)η_{k}.

Corollary 1. For m ∈ ℕ and α ∈ ℝ \ ℚ, the maximum length, d_{α}(m), and minimum length, d′_{α}(m), of the subintervals of partition P_{α}(m), are given by:

(a) When q_{k} > s + 1,

d_{α}(m) = \{ \begin{matrix}η_{k+1} + η_{k} & , r = a_{k+1} \\ η_{k+1} + (a_{k+1} − r + 1)η_{k} & , r < a_{k+1} \end{matrix}

When q_{k} = s + 1,

d_{α}(m) = \{ \begin{matrix}η_{k} & , r = a_{k+1} \\ η_{k+1} + (a_{k+1} − r)η_{k} & , r < a_{k+1} \end{matrix}

(b) For all q_{k} ≥ s + 1,

d′_{α}(m) = \{ \begin{matrix}η_{k+1} & , r = a_{k+1} \\ η_{k} & , r < a_{k+1} \end{matrix}

where k, r, s, a_{k}, and η_{k} are as in Theorem 1.

Proof. From Remark 1(a) and Lemma 1 we have,

η_{k−1} − rη_{k} = \{ \begin{matrix}η_{k+1} & < & η_{k} , r = a_{k+1} \\ η_{k+1} + (a_{k+1} − r)η_{k} & > & η_{k} , r < a_{k+1} \end{matrix}

Now, the corollary follows from Theorem 1, Remark 1(b) and Remark 1(c). □

Theorem 2 (Main Theorem). Let α ∈ ℝ \ ℚ, m ∈ ℕ, and let d_{α}(m), d′_{α}(m) be, respectively, the maximum and minimum lengths of the subintervals of the partition

P^{α}(m). The sequence (\frac{d_{α}(m)}{d′_{α}(m)})_{m=1}^{∞} is bounded if and only if α is an irrational

number of constant type.

Proof. Let m = rq_{k} + q_{k−1} + s, where k, r, and s are the unique integers given by Theorem 1. From Corollary 1 and Lemma 1, we have

\frac{d_{α}(m)}{d′_{α}(m)} = \{ \begin{matrix}ϵ + \frac{η_{k+2}}{η_{k+1}} + a_{k+2} & , r = a_{k+1} \\ ϵ + \frac{η_{k+1}}{η_{k}} + (a_{k+1} − r) & , r < a_{k+1} \end{matrix}

where ϵ = 1 for q_{k} > s + 1 and ϵ = 0 for q_{k} = s + 1.

(a) If α is of constant type (Definition 1), then the partial quotients, a_{n}, of α, satisfy a_{n} ≤ B(α) < ∞ for all n ≥ 1. Since \frac{η_{j+1}}{η_{j}} < 1 for all j ≥ 0 (by Lemma 1),

\frac{d_{α}(m)}{d′_{α}(m)} < B(α) + 2 for all m ∈ ℕ. Hence, (\frac{d_{α}(m)}{d′_{α}(m)})_{m=1}^{∞} is bounded.

(b) Suppose \frac{d_{α}(m)}{d′_{α}(m)} < B_{0} where 0 < B_{0} < ∞ for all m ∈ ℕ. In particular, for m =

q_{k+1} [corresponding to r = a_{k+1}, s = 0], we have \frac{d_{α}(q_{k+1})}{d′_{α}(q_{k+1})} = ϵ + \frac{η_{k+2}}{η_{k+1}} + a_{k+2} < B_{0}

for all k ≥ 0. Hence, a_{k+2} < B_{0} for all k ≥ 0. Setting B = max{B_{0}, a_{1}}, we have a_{n} ≤ B for all n ≥ 1, and hence α is of constant type. □

Acknowledgments

We would like to thank Professor Paul Zweifel, Virginia Tech, for his encouragement and stimulating questions, which led, in part, to the present work. We would also like to thank Robin Endelman, Department of Mathematics, Virginia Tech, for many helpful suggestions and discussions.

References

[1] S. Drobot, Real Numbers, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964

[2] G. Karner, On quantum twist maps and spectral properties of Floquet operators, Ann. Inst. H. Poincaré A, 68 (1998), to appear.

[3] J. Shallit, Real numbers with bounded partial quotients: a survey, Enseign. Math., 38 (1992), 151–187.

[4] N. B. Slater, Gaps and steps for the sequence nθ mod 1, Proc. Camb. Phil. Soc., 63 (1967), 1115–1123.

Mathematical Physics Group, Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 USA

Current address: Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011 USA

manash@physics.uc.edu http://www.physics.uc.edu/˜ manash/

Institut für Kerntechnik und Reaktorsicherheit, Universität Karlsruhe (TH), Postfach 3640, D-76021 Karlsruhe, Germany

karner@irs.fzk.de

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-3.html.