New York Journal of Mathematics

New York J. Math. 14 (2008) 139–143.

The relative growth rate for partial quotients

Andrew Haas

Abstract. The rate of growth of the partial quotients of an irrational number is studied relative to the rate of approximation of the number by its convergents. The focus is on the Hausdorff dimension of exceptional sets on which different growth rates are achieved.

In this note we look at the rate of growth of the partial quotients a_{i} of the irrational number

x = [a_{1}, a_{2}, …] = \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + ⋯}}}

relative to the rate at which x is approximated by its rational convergents. For x ∈ (0, 1) irrational, let \{\frac{p_{n}}{q_{n}} \} be the sequence of rational convergents given by the continued fraction expansion of x [7]. It follows from classical results of Khinchin and Lévy [3, 5] that for almost all x

(1)  lim_{n→∞} \frac{log a_{n}}{n} = 0 and lim_{n→∞} −\frac{log |x − \frac{p_{n}}{q_{n}} |}{n} = \frac{π^{2}}{6 log 2}.

Consequently, for almost all x ∈ (0, 1)

(2)  lim_{n→∞} \frac{log a_{n+1}}{log |x − \frac{p_{n}}{q_{n}} |} = 0.

Here we study the Hausdorff dimension of exceptional sets on which the limit (2) either does not exist or is different from zero. Similar, nonoverlapping, problems are considered in [9] using more sophisticated methods of multifractal analysis. The original motivation for looking at this question

139^{ISSN 1076-9803/08}

Received November 12, 2007.

Mathematics Subject Classification. 11K50, 11K60.

Key words and phrases. metric diophantine approximation, Hausdorff dimension.

arose in our study of another means for describing the rate of growth of partial quotients [6].

We shall write Dim_{H}X for the Hausdorff dimension of a set X ⊂ [0, 1] and H^{s}(X) for the Hausdorff s-dimensional measure of X [4].

Let

(3)  ℱ (z) = \{x ∈ (0, 1) : lim sup_{n→∞} \frac{− log a_{n+1}}{log |x − \frac{p_{n}}{q_{n}} |} = z\} .

Then we have:

Theorem 1. For 0 < z ≤ 1, Dim_{H}ℱ (z) = 1 − z and H^{1−z}(ℱ (z)) = ∞. If z ∉ [0, 1] then ℱ (z) = ∅.

By an earlier remark, ℱ (0) is a set of Lebesgue measure 1.

There is an alternative characterization of the problem in terms that compare the rate of growth of the denominators of the convergents to the rate at which they approximate x. It is, in its own right an interesting way to look at the problem. For α ∈ ℝ define the set

𝒢 (α) = \{x ∈ (0, 1) : lim sup_{n→∞} \frac{log q_{n}^{2}}{log |x − \frac{p_{n}}{q_{n}} |} = α\} .

Lemma 1. When z ∈ [0, 1], ℱ (z) = 𝒢 (z − 1).

Proof. Define the approximation constants θ_{n}(x) = q_{n}|q_{n}x − p_{n}|. From the classical theory of continued fractions we have

(4)  θ_{n}(x) = \frac{1}{a′_{n+1} + \frac{q_{n−1}}{q_{n}}}

where a′_{n+1} = a_{n+1} + [a_{n+2}, …] [7]. Therefore

lim sup_{n→∞} \frac{− log a_{n+1}}{log |x − \frac{p_{n}}{q_{n}} |} = lim sup_{n→∞} \frac{log θ_{n}(x)}{log |x − \frac{p_{n}}{q_{n}} |} = lim sup_{n→∞} \frac{log q_{n}^{2}}{log |x − \frac{p_{n}}{q_{n}} |} + 1. □

At this point it is an easy matter to show that ℱ (1) is an infinite set and therefore H^{0}(ℱ (1)) = ∞. In fact, if one chooses the partial quotients so that q_{n}^{2n} < a_{n+1}, then using (4) it follows that the limit in (2) is equal to 1.

In order to complete the proof of Theorem 1, we shall work with the alternative formulation suggested by the lemma and prove:

Theorem 2. For α ∈ (−1, 0], Dim_{H}𝒢 (α) = |α| and H^{|α|}(𝒢 (α)) = ∞. If α ∉ [−1, 0], then 𝒢 (α) = ∅.

The set 𝒢 (−1) = ℱ (0) has Lebesgue measure 1. Interestingly, the results in [9] imply that if the second limit in (1) exists for a number x (not necessarily taking the value given in (1)), then x ∈ 𝒢 (−1).

The last sentence of Theorem 2 is elementary and is a consequence of the following basic property of the convergents [7]

|x − \frac{p_{n}}{q_{n}} | < \frac{1}{q_{n}^{2}} .

The main tool in the proof of Theorem 2 is Jarník’s “zero-infinity” law [1, 2, 8]. We need to establish some notation and reframe the problem so that Jarník’s Theorem will apply.

The abbreviation FIM will be used in place of the phrase, “for infinitely many.” Given τ ∈ (−1, 0), and 0 ≤ ϵ < |τ|, define

ψ_{(τ,ϵ)}(r) = r^{\frac{2}{τ+ϵ}} .

Consider the related equation

(5)  |x − \frac{p}{q}| < ψ_{(τ,ϵ)}(q) = q^{\frac{2}{τ+ϵ}}

and the set

W (ψ_{(τ,ϵ)}) = \{x ∈ [0, 1] : |x − \frac{p}{q}| < q^{\frac{2}{τ+ϵ}} FIM \frac{p}{q} ∈ ℚ\} .

We are not interested in just any rationals but rather in the convergents. Define

W^{∗}(ψ_{(τ,ϵ)}) = \{x ∈ [0, 1] : |x − \frac{p_{n}}{q_{n}} | < qn^{\frac{2}{τ+ϵ}} FIM convergents \frac{p_{n}}{q_{n}} of x\} .

Observe that when q is sufficiently large,

(6)  q^{\frac{2}{τ+ϵ}} < \frac{1}{2}q^{−2}.

If \frac{p}{q} satisfies inequalities (5) and (6) then it is a convergent of x [7]. Therefore, except for finitely many rationals, \frac{p}{q} satisfies (5) if and only if it is a convergent of x. Consequently, W (ψ_{(τ,ϵ)}) = W^{∗}(ψ_{(τ,ϵ)}).

Combining the last observation with a simple manipulation of equation (5) yields

W (ψ_{(τ,ϵ)}) = \{x ∈ [0, 1] : \frac{log q_{n}^{2}}{log |x − \frac{p_{n}}{q_{n}} |} > τ + ϵ FIM convergents \frac{p_{n}}{q_{n}} of x\} .

It is therefore clear that for −1 < τ < α ≤ 0

(7)  W (ψ_{(τ,0)}) ⊃ 𝒢 (α).

Now we turn to the computation of Hausdorff dimension.

Lemma 2. For any τ ∈ (−1, 0), Dim_{H}W (ψ_{(τ,0)}) = |τ|, H^{|τ|}(W (ψ_{(τ,0)})) = ∞ and for ϵ > 0, H^{|τ|}(W (ψ_{(τ,ϵ)})) = 0.

Proof. Since ψ_{(τ,ϵ)} : ℝ^{+} → ℝ^{+} is a decreasing function, a basic version of Jarník’s Theorem [2] says that for s ∈ (0, 1)

H^{s}(W (ψ_{(τ,ϵ)})) = \begin{cases}0 & if ∑_{r=1}^{∞} r(ψ_{(τ,ϵ)}(r))^{s} < ∞ \\ ∞ & if ∑_{r=1}^{∞} r(ψ_{(τ,ϵ)}(r))^{s} = ∞. \end{cases}

The series involved are easy to analyze and it follows that for τ ∈ (−1, 0) and 0 ≤ ϵ < |τ|,

H^{s}(W (ψ_{(τ,ϵ)})) = \begin{cases}0 & for s > −τ − ϵ \\ ∞ & for s ≤ −τ − ϵ. \end{cases}

From this we conclude that Dim_{H}W (ψ_{(τ,0)}) = |τ| and moreover, that W (ψ_{(τ,0)}) has infinite |τ|-measure. Also, when ϵ > 0 the sets W (ψ_{(τ,ϵ)}) have |τ|-measure zero. □

Proof of Theorem 2. First, it follows from the inclusion (7) and Lemma 2 that

(8)  Dim_{H}𝒢 (α) ≤ Dim_{H}W (ψ_{(τ,0)}) = |τ|

for all −1 < τ < α ≤ 0. In particular, this gives Dim_{H}𝒢 (0) = 0.

Now suppose α ∈ (−1, 0) and pick k < 0 so that \frac{1}{k} < |α|. Define the set

E(α) = W (ψ_{(α,0)}) \ [ ⋃_{n=k}^{∞} W (ψ_{(α, \frac{1}{n})})] .

It is clear that E(α) ⊂ 𝒢 (α). Furthermore, applying Lemma 2, we see that Dim_{H}E(α) = |α| and H^{|α|}(E(α)) = ∞. Thus,

(9)  |α| = Dim_{H}E(α) ≤ Dim_{H}𝒢 (α).

Together equations (8) and (9) allow us to conclude that Dim_{H}𝒢 (α) = |α|. Since E(α) has infinite |α|-measure, so must the larger set 𝒢 (α). □

Acknowledgements. I am very grateful to the referee, whose feedback and advice resulted in a far stronger paper.

References

[1] Beresnevich, Victor; Dickinson, Detta; Velani, Sanju. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179 (2006), no. 846. MR2184760 (2007d:11086), Zbl pre05014118

[2] Beresnevich, Victor; Velani, Sanju. Ubiquity and general logarithm law for geodesics. arXiv:0707.1225v1.

[3] Billingsley, Patrick. Ergodic theory and information. J. Wiley & Sons, New York-London, 1965. MR0192027 (33 #254), Zbl 0141.16702

[4] Falconer, Kenneth. Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Ltd., Chichester, 1990. MR1102677 (92j:28008), Zbl 0689.28003

[5] Haas, Andrew. An ergodic sum related to the approximation by continued fractions. New York J. Math. 11 (2005), 345–349. MR2154360 (2006d:11080), Zbl pre02245713.

The relative growth rate for partial quotients 143

[6] Haas, Andrew. Geodesic cusp excursions and metric diophantine approximation. arXiv:0709.0313v1.

[7] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. 5th edition. The Clarendon Press, Oxford University Press, New York, 1979. MR0568909 (81i:10002), Zbl 0423.10001.

[8] Jarník, Vojtěch. Über die simultanen diophantischen Approximationen, Math. Z., 33 (1931), 505–543. MR1545226, Zbl 0001.32403, JFM 57.1370.01.

[9] Kesseböhmer, Marc; Stratmann, Bernd O. Multifractal analysis for Stern– Brocot intervals, continued fractions and Diophantine growth rates. J. Reine Angew. Math. 605 (2007), 133–163. MR2338129, Zbl pre05163325.

University of Connecticut, Department of Mathematics, Storrs, CT 06269 haas@math.uconn.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-5.html.