ESTIMATING THE FRACTAL DIMENSION OF SETS DETERMINED BY NONERGODIC PARAMETERS

Given fixed and irrational 0 < α, θ < 1, consider the billiard table Bα formed by a 1 2 ×1 rectangle with a horizontal barrier of length α emanating from the midpoint of a vertical side and a billiard flow with trajectory angle θ. In 1969, Veech introduced two subsets K0 (θ) and K1 (θ) of R/Z that are defined in terms of the continued fraction representation of θ ∈ R/Z, and Veech showed that these sets have Hausdorff dimension 0 when θ is rational. Moreover, the set K1 (θ) describes the set of all α such that the billiard flow on Bα in direction θ is nonergodic. We show that the Hausdorff dimension of the sets K0 (θ) and K1 (θ) can attain any value in [0, 1] by considering the continued fraction expansion of θ. This result resolves an analogue of work completed by Cheung, Hubert, and Pascal in which they consider, for fixed α, the set of θ such that the flow on Bα in direction θ is nonergodic.

1.1. A geometric interpretation of Veech's skew products and a physical interpretation of the set K 1 (θ). Veech [8] constructed examples of minimal and not uniquely ergodic dynamical systems as follows (see [6]). Take two copies of the unit circle and mark off a segment J of length 2πα in the counterclockwise direction on each one with endpoint at 0. Now choose an irrational θ and consider the following dynamical system. Start with a point p in the first circle. Rotate counterclockwise by 2πθ repeatedly until the the orbit lands in J; then switch to the corresponding point in the second circle, rotate by 2πθ until the first time the point lands in J; switch back to the first circle and so forth. Veech showed there exists irrational α for which this system is minimal and the Lebesgue measure is not uniquely ergodic. In this particular context, being not uniquely ergodic implies that there are open subsets of R/Z for which the amount of time the orbit spends in those open sets is not proportional to the sizes of the sets.
We may describe Veech's dynamical system using a flow on a surface arising from a billiard. Consider billiards in the table formed by a 1 2 × 1 rectangle with a horizontal barrier of length α with one end touching at the midpoint of a vertical side. We can identify the top half of the table as the positive side and the bottom half as the negative side. A standard unfolding of this billiard table is shown in Figure 2. We can view the unfolded table as having two (identified) barriers of length 2α.
It has been a difficult challenge in computing the Hausdorff dimension of the set N E all (α, θ) ∈ R 2 such that the billiard flow is non-ergodic. In an elegant work of Cheung, Hubert and Masur [4], the Hausdorff dimension of the set of all θ given α (i.e., the vertical slices of the set of non-ergodic directions), was computed as either 0 or 1 2 . Specifically, they showed that the dimension is 1 2 if and only if k∈N log log q k+1 q k < ∞ and α is irrational, where α = [0; q 1 , q 2 , . . .]. The goal of this paper is to consider for the first time the Hausdorff dimension of the horizontal slices of the set N E. In contrast to the dichotomy result for the vertical slices, we show that these Hausdorff dimensions can be any numbers between 0 and 1. Moreover, K 1 (θ) is the set of all α for which the flow in direction θ is nonergodic (see [3]).
1 From now on we omit the expression mod 1.  1.2. Motivation for considering θ divergent relative to some fixed integer M . In [8] Veech also showed that the sets K i (θ) have dimension 0 when sup a k < ∞. Hence, if we wish to construct continued fractions θ such that HdimK i (θ) > 0, then we must consider those values of θ satisfying sup a k = ∞. Further, in unpublished work by Lothar Narins [7], it was conjectured that the sets K i (θ) have dimension 1 when the terms a k satisfy n δ k < a k+1 < 2n δ k for some δ ∈ (0, 1) and sufficiently large k. We prove a weaker version of this conjecture in Section 4.2 by imposing additional constraints on the sequence (b k ) k∈N . Motivated by these observations, we amalgamate these two approaches in order to construct a class of continued fractions for which we can find upper and lower bound estimates for the dimension of K i (θ) other than 1 and 0, respectively. Specifically, for any 0 ≤ δ ≤ 1 and any integer M ≥ 3, we can construct a continued fraction θ divergent relative to M such that HdimK 0 (θ) ≤ δ and δ ≤ HdimK 1 (θ) by applying upper and lower bound formulas constructed in Section 2 and Section 4, respectively.
1.3. Sketch of the derivation of the upper and lower bound formulas for HdimK i (θ). In Section 2 we find an upper bound formula for HdimK 0 (θ) for all continued fractions θ that are divergent relative to some integer M . This is achieved by applying a construction from [2] known as a self-similar covering to members of a family of subsets K i 0 (θ) i∈N of K 0 (θ). The reason for reducing to a family of subsets is two-fold: first, it is much more convenient to apply the selfsimilar covering to the subsets K i 0 (θ) than to K 0 (θ), and second, we will show that HdimK 0 (θ) = sup i HdimK i 0 (θ) by showing that K 0 (θ) = K i 0 (θ) in Lemma 2.2 and exhibiting an upper bound for Hdim K i 0 (θ). We construct a lower bound for HdimK 1 (θ) on two separate occasions by applying a lower bound formula given by Falconer [5]. Falconer's inequality applies to Cantor sets satisfying some mild conditions, which are stated in Section 4.1, so some of our work is dedicated to constructing Cantor sets contained in K 1 (θ) that satisfy the conditions needed to apply the lower bound formula. In each occasion for which Falconer's formula is applied, we proceed by constructing an infinite family of Cantor sets contained in K 1 (θ) in a way that allows us to get lower bounds of the Cantor sets, and hence lower bounds of HdimK 1 (θ), arbitrarily close to some specified value.
1.4. Overview of the paper. In Section 2 we give an upper bound for HdimK 0 (θ) for θ divergent relative to any fixed M ∈ N. Section 3 is devoted to an application of the upper bound formula in which we construct a continued fraction θ such that HdimK 0 (θ) = HdimK 1 (θ) = 0. In Section 4 we apply Falconer's lower bound formula to give a lower bound for HdimK 1 (θ), and we use this lower bound to construct a continued fraction θ such that HdimK i (θ) = 1. We apply both upper and lower bound formulas in Section 5 to show that for any δ ∈ (0, 1) and M ∈ N ≥3 we can construct a continued fraction such that HdimK 0 (θ) = HdimK 1 (θ) = δ.

2.
Upper bound for Hausdorff dimension. In this section we give an upper bound formula for HdimK 0 (θ) by applying a self-similar covering, introduced by Cheung [2], to subsets of K 0 (θ), which we call K i 0 (θ) and define below. In particular, the self-similar covering will allow us to give an upper bound for HdimK i 0 (θ), and we show that this upper bound is also an upper bound for HdimK 0 (θ).
Definition. Let M ∈ N. An irrational θ with unbounded partial quotients is divergent relative to M if the subsequence of partial quotients formed by the terms that are greater than M diverges to ∞.
If θ is divergent relative to M , define The following theorem gives an upper bound formula for HdimK 0 (θ) when θ is divergent relative to M . Theorem 2.1. Let θ be divergent relative to M , and let (k i ) ∞ i=0 enumerate the numbers k satisfying a k+1 > M in increasing order. Then We devote the remainder of Section 2 to proving Theorem 2.1.

2.1.
Reduction to a family of subsets of K 0 (θ). In this section we construct a family of subsets of K 0 (θ). This will simplify our calculations and still allow us to obtain upper bounds of HdimK 0 (θ).
Definition. Suppose θ is divergent relative to M . Define The sets K i 0 (θ) are nonempty for all sufficiently large i since, by definition of K 0 (θ), we have lim j→∞ |b j | n j n j θ = 0.
Proof. For any set F ⊂ R and s ≥ 0, denote by H s (F ) the s-dimensional Hausdorff measure of F . Then HdimK i For each i we can cover K i 0 (θ) by intervals A ij such that the sum of their radii by the power s + ε is less than 2 −i δ. The union of all intervals A ij , over i and j, covers i K i 0 (θ) and the sum of their radii raised by the power s + ε is less than δ. Therefore, A consequence of the following lemma is that each x ∈ K i0 0 (θ) can be expressed as m; b 1 , b 2 , . . . θ such that for all j > k i0 either a j+1 > M or b j = 0. This phenomenon was the motivation for defining the sets K i 0 (θ). Lemma 2.4. There exists an integer i 0 such that if j ≥ k i0 , j ∈ κ θ and |b j | n j n j θ < 1 4M , then b j = 0. Proof. Take i 0 sufficiently large so that k i0 ≥ min κ θ . Since j ∈ κ θ , a j+1 ≤ M .

2.2.
Specification of self-similar covering. Given |b i | ≤ a i+1 and an even b satisfying |b| ≤ a k+1 , define by Since n k+2 > 2n k , we have 1 which is a geometric series that simplifies to 2 n k+1 . Similarly, Without loss of generality, suppose m = 0. Then Therefore, Definition. (Section 3 of [2]) Let X be a metric space and J a countable set. Given σ ⊂ J × J and α ∈ J we let σ (α) denote the set of all α ∈ J such that (α, α ) ∈ σ.
We say a sequence (α j ) j∈N of elements in J is σ-admissible if α j+1 ∈ σ (α j ) for all j ∈ N; and we let J σ denote the set of all σ-admissible sequences in J. By a selfsimilar covering of X we mean a triple (B, J, σ) where B is a collection of bounded subsets of X, J a countable index set for B, and σ ⊂ J × J such that there is a map E : K 0 (θ) → J σ that assigns to each x ∈ X a σ-admissible sequence α x j j∈N such that for all x ∈ X we have the following: where B (α) denotes the element of B indexed by α.

2.3.
A self-similar covering of K i0 0 (θ). We have access to a self-similar covering of K i0 0 (θ). Define Let J σ denote the set of all σ-admissible sequences in J. By Lemma 2.5 we can define 0 (θ). Our triple (B, J, σ) satisfies (i) of the definition of a self-similar covering; apply Lemma 2.5 to show x ∈ I (β j ) for all β j ∈ J 0 . Since lim j→∞ diam I α x j = 0, where the union is over all finite sequences m, b 1 , . . . , b ki−1 with k i ∈ κ θ and |b j | ≤ a j+1 for each j.

2.4.
Calculation. In this section we give an upper bound on HdimK i0 0 (θ). The following lemma is a direct consequence of the definition of σ.
Lemma 2.6. Let α ∈ J and k = |α|.Then Proof. Let α be a sequence in J of length k i ∈ κ θ . Then α = (m; b 1 , . . . , b ki−1 ) where both j > k i0 and j ∈ κ θ imply b j = 0. Since |α| = k, α corresponds to an interval belonging to E i . Given an interval centered at mθ + j i=1 b i n i θ , let us call the intervals centered at mθ + j i=1 b i n i θ + b j+1 n j+1 θ with |b j | ≤ a j+1 the children intervals of the original interval. Hence, the centers determined by the children of the intervals comprising E i are determined by all even integers b k satisfying b k ≤ a ki+1 .
Since the value of |I (β)| does not depend on the choice of β ∈ σ (α), a direct consequence of inequality 3 is that for each s ≥ 0 we have Further, if β ∈ σ (α), then The critical value of s is s = log #σ(α) log n k i+1 −log n k i . Let ε > 0, and let s = s + ε. Then
We define x := max {n ∈ Z : n ≤ x}, x := min {n ∈ Z : n ≥ x}, and denote by [x] the nearest integer to x (to avoid ambiguity when rounding half-integers, we round to the nearest even integer). Let 0 < δ ≤ 1 be given and fix M = 2 1 δ . We specify the continued fraction representation of θ recursively. We choose a 1 , a 2 , . . . , a k0 = M , where k 0 is the smallest index for which n δ k0 > max 2M, 1 2 δ −1 .
Since θ is divergent relative to M ,

4.
A Lower bound for HdimK 1 (θ). In Section 4.1 we state a lower bound formula given by Falconer [5] for a particular class of Cantor sets. We can approximate K 1 (θ) by a family of Cantor sets, so we show how Falconer's formula can be used to give a lower bound for K 1 (θ). In Section 4.2 we use Falconer's formula to provide an example of a continued fraction θ such that HdimK 1 (θ) = 1.
be a decreasing sequence of sets, with each F k a union of a finite number of disjoint closed intervals (called kth level basic intervals), with each interval of F k containing at least two intervals of F k+1 , and the maximum length of kth level intervals tending to 0 as k → ∞. Then the set The condition needed to apply Falconer's lower bound formula is ( * ) For j ∈ N, each (j − 1)th level interval contains at least m j ≥ 2 jth level intervals that are separated by gaps of at least γ j , where 0 < γ j+1 < γ j for each j.
Falconer's lower bound formula is 4.
2. An example of θ such that HdimK 1 (θ) = 1. In this section we apply inequality 8 to construct a continued fraction θ such that HdimK 1 (θ) = 1. Fix δ ∈ (0, 1) and ε ∈ (0, δ). For the remainder of Section 4, let θ be an element such that there exists a k 0 sufficiently large so that n δ−ε k0 ≥ 3, n ε k0 ≥ 9, and for all k ≥ k 0 , n δ k < a k+1 < 2n δ k , b k is even and |b k | < n δ−ε k . Our strategy is to construct an infinite family of Cantor sets contained in K 1 (θ), each of which satisfies the conditions ( * ) needed to apply Falconer's lower bound formula, allowing us to give lower bounds of HdimK 1 (θ) arbitrarily close to 1.
Given Proof. The distance between the centers of adjacent intervals L (m; b 1 , b 2 , . . . , b k+1 ) and L (m; b 1 , b 2 , . . . , b k+1 + 2) is 2 n k θ . Between these centers is the gap between the intervals as well as two half intervals (specifically, the right half of L (m; b 1 , b 2 , . . . , b k+1 ) and the left half of L (m; b 1 , b 2 , . . . , b k+1 + 2)). Hence, the size of the gap is n k θ .

Define
M j := a ε kj +1 , so that M j is a lower bound on the number of intervals in F j+1 contained in intervals of F j . Define Γ j := n kj θ , so that, by Lemma 4.1, Γ j is a lower bound on the gaps between the intervals of F j+1 . Since 1 n k j+2 < 1 2n k j+1 for each j, we have 0 < Γ j+1 < Γ j . Inequality (9)