Asymptotic estimates of large gaps between directions in certain planar quasicrystals

Given a locally finite point set ${\mathcal{P}}$ in the plane $\mathbb{R}^{2}$, we consider its set of directions, that is, the set of points $\widehat{{\text{\boldmath$v$}}}:=\|{\text{\boldmath$v$}}\|^{-1}{\text{\boldmath$v$}}$ on the unit circle $\operatorname{S{}}_{1}^{1}$ as $v$ runs through ${\mathcal{P}}\smallsetminus\{\mathbf{0}\}$. For each $R>0$, let $\Delta_{R}$ be the finite multi-set of directions to points in ${\mathcal{P}}$ within distance $R$ from the origin, i.e.,

Throughout the paper we will assume that ${\mathcal{P}}$ has an asymptotic density $c_{\mathcal{P}}>0$, meaning that for any bounded set ${\mathcal{D}}\subset\mathbb{R}^{2}$ with boundary of measure zero,

It then follows that the multi-set $\Delta_{R}$ has cardinality $\sim c_{\mathcal{P}}\hskip 1.0pt\pi R^{2}$ as $R\to\infty$, and furthermore that this multi-set becomes asymptotically equidistributed along the unit circle, in the sense that for any arc $I\subset\operatorname{S{}}_{1}^{1}$,

where $|I|$ denotes the length of $I$ (in particular $|\operatorname{S{}}_{1}^{1}|=2\pi$).

In this situation, it is natural to consider finer questions about the local statistics of the points in $\Delta_{R}$. A particular statistics which has been much studied is the distribution of normalized gaps between the points in $\Delta_{R}$, as $R\to\infty$. For example, when ${\mathcal{P}}$ is the set of primitive lattice points in $\mathbb{Z}^{2}$, the limiting distribution of normalized gaps was explicitly determined by Boca, Cobeli and Zaharescu [9]. More generally, if ${\mathcal{P}}$ is an arbitrary lattice (possibly translated) in $\mathbb{R}^{d}$, for any $d\geq 2$, the limiting distribution of general local statistics of directions was proved to exist by Marklof and Strömbergsson [25]; see also [14] and [23] regarding convergence of related moments. In particular it was noted in [25] that the explicit limiting distribution computed in [9] remains valid when replacing $\mathbb{Z}^{2}$ by an arbitrary lattice in $\mathbb{R}^{2}$; furthermore, when ${\mathcal{P}}$ is an arbitrary ’irrational’ translate of a lattice in $\mathbb{R}^{2}$, the limiting distribution of gaps between directions coincides with the gap distribution for the fractional parts of $\sqrt{n}$ calculated by Elkies and McMullen [15]; see [25, Sec. 1.2].

Athreya and Chaika [2, Prop. 3.10] have proved that the limiting distribution of gaps between directions exists in the case when ${\mathcal{P}}$ is the set of holonomy vectors of either the saddle connections or periodic cylinders on a generic translation surface; see also [3], [44], [24], [40], [8] and [35] for later studies with more precise results in specific cases. Also in hyperbolic $n$-space, for any point set which is the orbit of a lattice within the group of orientation preserving isometries, the analogous limiting gap distribution and also more general local statistics of directions, has been proved to exist by Marklof and Vinogradov [31]; for related work see [10], [22], [38].

The present paper concerns the case of ${\mathcal{P}}$ being a regular cut-and-project set (also referred to as a Euclidean model set). For this case, the limit distribution of normalized gaps between directions was proved to exist, and given a complicated but explicit description, by Marklof and Strömbergsson in [27]; see Theorem 1 below. See also Rühr, Smilansky and Weiss [39] for closely related investigations, and El-Baz [13] for an extension to so called adelic model sets. The results of [27] answered some questions which had been raised by Baake, Götze, Huck and Jakobi in [4], where a numerical investigation was carried out of the normalized gap distribution between directions for several vertex sets coming from aperiodic tilings, some of these being of cut-and-project type and others not. Building on the results of [27], Hammarhjelm [20] explicitly determined the limit of the minimal normalized gap between the directions to the points in ${\mathcal{P}}$, for ${\mathcal{P}}$ belonging to either of two families of planar quasicrystals, including both the Ammann-Beenker point set and the vertex sets of some rhombic Penrose tilings. Also in [20], the asymptotic density of visible points was explicitly determined for several families of planar quasicrystals, including the two families just mentioned.

The main purpose of the present paper is to continue the explicit study begun in [20] of the limit distribution of normalized gaps between directions in the case of ${\mathcal{P}}$ belonging to certain families of planar cut-and-project sets. Our focus will be on asymptotics for large gaps between directions. The present work grew out of the initial study carried out by Hammarhjelm [19].

We remark that the methods developed in the present paper can be expected to be useful also for questions related to the Lorentz gas on a cut-and-project scatterer configuration – namely, for the task of obtaining asymptotic estimates for the transition probabilities in the transport (Markov) process which arises when considering such a Lorentz gas in the limit of low scatterer density [26], [29]. For the case of a lattice scatterer configuration, such asymptotic estimates were obtained in [28], and found an important application in [30].

To recall the precise definition of cut-and-project sets considered in [27], let $d,m\geq 1$, set $n=d+m$, and let ${\mathcal{L}}$ be a lattice of full rank in $\mathbb{R}^{n}=\mathbb{R}^{d}\times\mathbb{R}^{m}$. We refer to $\mathbb{R}^{d}$ and $\mathbb{R}^{m}$ as the physical space and internal space, respectively. In the present paper the dimension of the physical space will always be $d=2$. We write $\pi$ and $\pi_{\operatorname{int}}$ for the orthogonal projection of $\mathbb{R}^{n}$ onto the first $d$ coordinates and last $m$ coordinates. Let ${\mathcal{A}}$ be the closure of $\pi_{\operatorname{int}}({\mathcal{L}})$; this is a closed abelian subgroup of $\mathbb{R}^{m}$. Let ${\mathcal{A}}^{\circ}$ be the identity component of ${\mathcal{A}}$; this is a linear subspace of $\mathbb{R}^{m}$, and set $m^{\prime}=\dim{\mathcal{A}}=\dim{\mathcal{A}}^{\circ}$. Let ${\mathcal{W}}$, the window, be a bounded subset of ${\mathcal{A}}$ with nonempty interior (with respect to the topology of ${\mathcal{A}}$). We will always assume that ${\mathcal{W}}$ and ${\mathcal{L}}$ are such that

We now define the cut-and-project set associated to ${\mathcal{W}}$ and ${\mathcal{L}}$ to be

We will always assume that ${\mathcal{P}}({\mathcal{W}},{\mathcal{L}})$ is regular, meaning that $\partial{\mathcal{W}}$ has measure zero with respect to the Haar measure of ${\mathcal{A}}$. Under these assumptions, it is known that ${\mathcal{P}}$ has an asymptotic density, i.e. (1.2) holds, with $c_{\mathcal{P}}$ being given by a simple explicit expression in terms of ${\mathcal{L}}$ and ${\mathcal{W}}$ [26, Prop. 3.2]; see also (1.11) below.

Let us view the unit circle $\operatorname{S{}}_{1}^{1}$ as the set of $z\in\mathbb{C}$ with $|z|=1$, and for any $z\in\operatorname{S{}}_{1}^{1}$ let us call the number $\frac{1}{2\pi}\arg(z)$ the normalized angle of $z$; this gives an identification between $\operatorname{S{}}_{1}^{1}$ and $\mathbb{R}/\mathbb{Z}$. Let us order the normalized angles of the points in $\Delta_{R}$ in an increasing list as

where $N(R)=\#\Delta_{R}$. Also set $\xi_{R,0}=\xi_{R,N(R)}-1$.

The following result was proved in [27] (see also Remark 1.7 below).

We recall the explicit formula for $F(s)$ from [27] in Section 3.3 below. Figure 1 shows conjectural graphs of the function $-F^{\prime}(s)$, i.e. the limiting density of normalized gaps, for some examples of vertex sets of Ammann-Beenker, Gähler’s shield, and Tübingen triangle tilings.

Our main goal in the present paper is to describe the tail asymptotics of the distribution function $F(s)$ in Theorem 1, for a particular class of cut-and-project sets in $\mathbb{R}^{2}$, which includes several classical examples. We expect that the methods which we develop can be extended to more general cases as well (in particular see Remark 1.33 below). We now give the description of the class which we will consider. Let $K$ be a real quadratic field; let ${\mathcal{O}}_{K}$ be its ring of integers; let $\sigma$ be the unique non-trivial automorphism of $K$, and let $\lambda\in\mathcal{O}_{K}^{\times}$ be the fundamental unit (thus $\lambda>1$ and $\mathcal{O}_{K}^{\times}=\left\{\pm\lambda^{n}\>:\>n\in\mathbb{Z}\right\}$). Let ${\mathcal{L}}_{K}$ be the Minkowski embedding of ${\mathcal{O}}_{K}^{2}$ in $\mathbb{R}^{4}$, viz.,

We set $d=m=2$, that is, we view $\mathbb{R}^{4}$ as the product of a 2-dimensional physical space and a 2-dimensional internal space, and $\pi$ and $\pi_{\operatorname{int}}$ denote the orthogonal projection of $\mathbb{R}^{4}$ onto the first 2 coordinates and the last 2 coordinates, respectively. Note that in this case $\pi_{\operatorname{int}}({\mathcal{L}}_{K})$ is dense in $\mathbb{R}^{2}$, i.e. we have ${\mathcal{A}}=\mathbb{R}^{2}$, and so we take the window ${\mathcal{W}}$ to be a bounded subset of $\mathbb{R}^{2}$ with nonempty interior. (In the present case the restriction of $\pi$ to ${\mathcal{L}}_{K}$ is injective, and so the property (1.3) is automatically fulfilled.) We will always assume that ${\mathcal{P}}:={\mathcal{P}}({\mathcal{W}},{\mathcal{L}}_{K})$ is regular, i.e. that $\partial{\mathcal{W}}$ has Lebesgue measure zero.

For this class of cut-and-project sets, the formula for the asymptotic density of ${\mathcal{P}}$, [26, (1.7)], becomes (see also Section 2 below):

where $\Delta_{K}$ is the discriminant of $K$.

As we prove in Remark 3.29 below, the limiting gap distribution function is unchanged if $\mathcal{W}$ is modified by any measure zero set. In particular, without loss of generality we may assume $\mathcal{W}$ is open (indeed, otherwise replace $\mathcal{W}$ by its interior). Finally, in the present paper we will make the key assumption that $\mathbf{0}\in{\mathcal{W}}$.

In our approach we actually work with the integral of the distribution function $F$, i.e. with

As we will note below (see Lemma 3.2), using the fact that $F$ is decreasing, an easy interpolation argument allows us to deduce an asymptotic formula for $F$ once we know an asymptotic formula for $G$.

Our main result is the following:

As we will now describe, after imposing one more assumption on the window set, we can further evaluate the aforementioned formula for $a_{\mathcal{P}}$. To state our result, we need to introduce some more notation. For $\theta\in\mathbb{R}$ we write $\text{k}_{\theta}:=\left(\begin{smallmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{R})$. Then for any subset $\mathcal{W}\subset\mathbb{R}^{2}$ we let $\ell_{\mathcal{W}}(\theta)$ be the projection of $\mathcal{W}\text{k}_{-\theta}$ on the $x$-axis. On top of the assumptions in Theorem 2, we will now assume that for each $\theta$, $\ell_{\mathcal{W}}(\theta)$ is an interval. Note that this assumption is always satisfied if ${\mathcal{W}}$ is connected, but it also holds for many non-connected window sets ${\mathcal{W}}$.

Note that since $\mathcal{W}$ is open and $\mathbf{0}\in\mathcal{W}$, $\ell_{\mathcal{W}}(\theta)$ is an open subset of $\mathbb{R}$ containing $0$. Hence if $\ell_{\mathcal{W}}(\theta)$ is assumed to be an interval, then we can parametrize it by two positive numbers $r(\theta),\nu(\theta)$ via

We next illustrate how our results apply in three cases of well-known planar quasicrystals. The three propositions below are all proved in Section 4.4. More details, and comparison with numerics, is provided in Section 5.

We first consider the Ammann-Beenker tiling, which was discovered by Robert Ammann in the 70s and first described in [17] and [1]. Specifically, we consider the “A5 set, variant (b)”, in the notation of [1]; see Figure 2 (top left panel) above for a small patch of this tiling. It is well-known that the set of vertices of an arbitrary Ammann-Beenker tiling can be generated using the cut-and-project construction. Specifically, let $K=\mathbb{Q}(\sqrt{2})$, and let ${\mathcal{W}}_{\operatorname{AB}}\subset\mathbb{R}^{2}$ be the open regular octagon centered at the origin of edge length $1$, oriented so that four of the edges are perpendicular to a coordinate axis. For each ${\text{\boldmath$w$}}\in\mathbb{R}^{2}$ we set

where $g_{1}:=\left(\begin{smallmatrix}1&0\\ 1&\sqrt{2}\end{smallmatrix}\right)$ and $g_{2}:=\left(\begin{smallmatrix}1&0\\ 1/\sqrt{2}\>&1/\sqrt{2}\end{smallmatrix}\right)$. Then for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{AB}}$ with the property that $\pi_{\operatorname{int}}({\mathcal{L}}_{K})\cap\partial{\mathcal{W}}^{(\operatorname{AB})}_{{\text{\boldmath$w$}}}=\emptyset$, the cut-and-project set ${\mathcal{P}}_{\operatorname{AB},{\text{\boldmath$w$}}}$ is the vertex set of an Ammann-Beenker tiling with one vertex at $\mathbf{0}$, and conversely, the vertex set of any Ammann-Beenker tiling having a vertex at $\mathbf{0}$ is (up to scaling and rotation) either equal to such a point set ${\mathcal{P}}_{\operatorname{AB},{\text{\boldmath$w$}}}$, or can be obtained as a limit of such point sets ${\mathcal{P}}_{\operatorname{AB},{\text{\boldmath$w$}}}$ in an appropriate topology. (See [5, Ch. 7.3] and Section 5.1 below.)

Note that because of (1.23) and the last statement in Theorem 1, the formula (1.14) in Theorem 2 applies to the cut-and-project set ${\mathcal{P}}_{\operatorname{AB},{\text{\boldmath$w$}}}$ for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{AB}}$. Here the constant $a_{{\mathcal{P}}}$ is given by a quite simple formula, also valid for much more general window sets:

Note that Proposition 1.1 applies when ${\mathcal{W}}={\mathcal{W}}^{(\operatorname{AB})}_{{\text{\boldmath$w$}}}$ for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{AB}}$, and the formula (1.25) holds whenever $w$ lies sufficiently near $\mathbf{0}$. In particular, (1.25) implies that

In Section 5.4, we present a comparison, for a few examples of points ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{AB}}$, between the exact values of $a_{{\mathcal{P}}}$ computed using Proposition 1.1, and numerically computed approximate values of $s^{2}F(s)$ for large $s$.

Next we consider Gähler’s shield tiling, which was discovered in [18]; these tilings are built up of a certain triple of tiles (a triangle, a square, and a hexagon called a ’shield’), equipped with local matching rules. The vertex set of a Gähler’s shield tiling can be obtained using the cut-and-project construction in the following way: Let $K=\mathbb{Q}(\sqrt{3})$, and let ${\mathcal{W}}_{\operatorname{Gh}}\subset\mathbb{R}^{2}$ be the open regular dodecagon centered at the origin of edge length $1$, so that four of the edges are perpendicular to a coordinate axis. For each ${\text{\boldmath$w$}}\in\mathbb{R}^{2}$ we set

where $g_{1}:=\left(\begin{smallmatrix}1&0\\ \sqrt{3}&2\end{smallmatrix}\right)$ and $g_{2}:=\left(\begin{smallmatrix}1&0\\ \sqrt{3}/2&1/2\end{smallmatrix}\right)$. Then for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{Gh}}$ with the property that $\pi_{\operatorname{int}}({\mathcal{L}}_{K})\cap\partial{\mathcal{W}}^{(\operatorname{Gh})}_{{\text{\boldmath$w$}}}=\emptyset$, the cut-and-project set ${\mathcal{P}}_{\operatorname{Gh},{\text{\boldmath$w$}}}$ is the vertex set of a Gähler’s shield tiling with one vertex at $\mathbf{0}$. (See [5, Ch. 7.3] and Section 5.2 below.)

Again, (1.14) in Theorem 2 applies to ${\mathcal{P}}_{\operatorname{Gh},{\text{\boldmath$w$}}}$ for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{Gh}}$, and regarding $a_{{\mathcal{P}}}$ we have:

Note that Proposition 1.2 applies when ${\mathcal{W}}={\mathcal{W}}_{{\text{\boldmath$w$}}}^{(\operatorname{Gh})}$ for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{Gh}}$, and the formula (1.29) holds whenever $w$ lies sufficiently near $\mathbf{0}$. Again see Section 5.4 for numerical computations related to the values of $a_{{\mathcal{P}}}$ for ${\mathcal{P}}={\mathcal{P}}_{\operatorname{Gh},{\text{\boldmath$w$}}}$.

Finally, we consider the Tübingen triangle tiling, which was discovered and studied in [6]; these tilings are built up of a certain pair of isosceles triangles. The set of vertices of a Tübingen triangle tiling can be obtained using the cut-and-project construction in the following way. Let $K=\mathbb{Q}(\sqrt{5})$ and $\tau=\frac{1}{2}(1+\sqrt{5})$, and let ${\mathcal{W}}_{\operatorname{\hskip 1.0ptTT}}\subset\mathbb{R}^{2}$ be the open regular decagon centered at the origin of edge length $\sqrt{(2+\tau)/5}$, oriented so that two of the edges are perpendicular to the first coordinate axis. For each ${\text{\boldmath$w$}}\in\mathbb{R}^{2}$ we set

where $g_{1}:=\left(\begin{smallmatrix}1&0\\ 0&\sqrt{(2+\tau)/5}\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\ \tau&2\end{smallmatrix}\right)$ and $g_{2}:=\left(\begin{smallmatrix}1&0\\ (\tau-1)/2\>&\sqrt{2+\tau}/2\end{smallmatrix}\right)$. Then for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{\hskip 1.0ptTT}}$ with the property that $\frac{1}{5}\pi_{\operatorname{int}}({\mathcal{L}}_{K})\cap\partial{\mathcal{W}}^{(\operatorname{\hskip 1.0ptTT})}_{{\text{\boldmath$w$}}}=\emptyset$, the cut-and-project set ${\mathcal{P}}_{\operatorname{\hskip 1.0ptTT},{\text{\boldmath$w$}}}$ is the vertex set of a Tübingen triangle tiling with one vertex at $\mathbf{0}$. (See [5, Ch. 7.3], [6, Sec. 4] and Section 5.3 below.)

Again, (1.14) in Theorem 2 applies to ${\mathcal{P}}_{\operatorname{\hskip 1.0ptTT},{\text{\boldmath$w$}}}$ for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{\hskip 1.0ptTT}}$, and regarding $a_{{\mathcal{P}}}$ we have:

Proposition 1.3 applies when ${\mathcal{W}}={\mathcal{W}}_{{\text{\boldmath$w$}}}^{(\operatorname{\hskip 1.0ptTT})}$ for any ${\text{\boldmath$w$}}\in{\mathcal{W}}_{\operatorname{\hskip 1.0ptTT}}$, and the formula (1.32) holds whenever $w$ lies sufficiently near $\mathbf{0}$. Again see Section 5.4 for related numerical computations.

In this section we give a brief review on backgrounds on real quadratic fields. Let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d\in\mathbb{N}$ square-free. Let $\mathcal{O}_{K}=\mathbb{Z}[\tau]\subset K$ be its ring of integers, where

Let $J_{K}$ be the (abelian) group of fractional ideals of $\mathcal{O}_{K}$, and let $P_{K}$ be the subgroup of principal ideals. We let $C_{K}=J_{K}/P_{K}$ be the ideal class group of $K$, and we call elements in $C_{K}$ ideal classes of $K$. For any nonzero $a_{1},\ldots,a_{m}\in K$, we denote by $\langle a_{1},\ldots,a_{m}\rangle$ the fractional ideal generated by $a_{1},\ldots,a_{m}$.

Let $\sigma$ be the unique non-trivial automorphism of $K$ and consider the embedding

We note that $\iota(I)$ is a lattice in $\mathbb{R}^{2}$ for any $I\in J_{K}$. In particular, $\iota(\mathcal{O}_{K})$ is the lattice generated by $(1,1)$ and $(\tau,\sigma(\tau))$, which is of covolume $\Delta_{K}^{1/2}$, where

is the discriminant of $K$. For any fractional ideal $I\in J_{K}$, its absolute norm is defined by

Note that Nr is multiplicative in $J_{K}$, that is, $\text{Nr}(I_{1}I_{2})=\text{Nr}(I_{1})\text{Nr}(I_{2})$ for any $I_{1},I_{2}\in J_{K}$. Moreover, for any $\alpha\in K\setminus\{0\}$, we have $\text{Nr}(\alpha I)=|\mathrm{N}(\alpha)|\text{Nr}(I)$, where $\mathrm{N}:K\to\mathbb{R}$ is the standard norm on $K$ given by $\mathrm{N}(\alpha):=\alpha\sigma(\alpha)$.

Let $\mathbb{H}$ be the upper half plane. The group $\operatorname{SL}_{2}(\mathbb{R})$ acts on $\mathbb{H}$ via the Möbius transformation: $g\cdot z=\frac{az+b}{cz+d}$ for any $g=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{R})$ and $z\in\mathbb{H}$. Let $\mathrm{H}=\operatorname{SL}_{2}(\mathbb{R})\times\operatorname{SL}_{2}(\mathbb{R})$ (which we view as a subgroup of $\operatorname{SL}_{4}(\mathbb{R})$ via the block diagonal embedding) and let $\mathbb{H}^{2}$ be the product of two upper half planes. The group $\mathrm{H}$ naturally acts on $\mathbb{H}^{2}$ via

Denote by $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\infty\}$ which can be identified with the boundary of $\mathbb{H}$. The action of $\mathrm{H}$ on $\mathbb{H}^{2}$ naturally extends to $\overline{\mathbb{R}}^{2}$ via the same formula. We also denote by $\overline{K}:=K\cup\{\infty\}$. It embeds into $\overline{\mathbb{R}}^{2}$ via the natural extension of the embedding $\iota:K\to\mathbb{R}^{2}$ (by sending $\infty\in\overline{K}$ to $(\infty,\infty)\in\overline{\mathbb{R}}^{2}$). With slight abuse of notation, we also denote this embedding from $\overline{K}$ to $\overline{\mathbb{R}}^{2}$ by $\iota$. We will also write $\iota$ for the group homomorphism $\operatorname{SL}_{2}(K)\to\mathrm{H}$ given by

The Hilbert modular group for the field $K$ is given by

We write $\Gamma_{K}$ for the embedding of $\operatorname{SL}_{2}({\mathcal{O}}_{K})$ in $\mathrm{H}$:

The discreteness of $\iota(\mathcal{O}_{K})$ in $\mathbb{R}^{2}$ implies that $\Gamma_{K}$ is a discrete subgroup of $\mathrm{H}$. Indeed, it is a non-uniform lattice in $\mathrm{H}$, that is, the homogeneous space $\Gamma_{K}\backslash\mathrm{H}$ is non-compact and has finite volume with respect to a Haar measure of $\mathrm{H}$. As a subgroup of $\mathrm{H}$, $\Gamma_{K}$ naturally acts on $\overline{\mathbb{R}}^{2}$, and this action preserves $\iota\bigl{(}\overline{K}\bigr{)}$. The orbits of $\iota\bigl{(}\overline{K}\bigr{)}$ under $\Gamma_{K}$ are called the cusps of $\Gamma_{K}$. We will often represent a cusp by an element in the corresponding $\Gamma_{K}$-orbit, or by an element in $\overline{K}$ (via the natural identification between $\iota(\overline{K})$ and $\overline{K}$). The following lemma, together with the obvious identification between $\iota(\overline{K})/\Gamma_{K}$ and $\overline{K}/\operatorname{SL}_{2}(\mathcal{O}_{K})$, shows that the number of cusps of $\Gamma_{K}$ equals the number of ideal classes of $K$.

Throughout the remainder of this paper, we fix $k_{1}=\infty,k_{2},\ldots,k_{\kappa}\in\overline{K}$ to be a complete list of cusps of $\Gamma_{K}$. Fix $a_{1}=1$, $a_{2},\ldots,a_{\kappa}\in{\mathcal{O}}_{K}$ and $c_{1}=0$, $c_{2},\ldots,c_{\kappa}\in{\mathcal{O}}_{K}$ so that $k_{i}=\frac{a_{i}}{c_{i}}$ for each $i$; then set $I_{i}:=\langle a_{i},c_{i}\rangle$. Then $I_{1}=\mathcal{O}_{K},I_{2},\ldots,I_{\kappa}$ are integral ideals which by Lemma 2.1 form a system of representatives of the ideal classes of $K$. (Note that $I_{i}$ depends on the choice of $a_{i},c_{i}$; however the class $[I_{i}]$ depends only on $k_{i}$.)

For each $1\leq i\leq\kappa$, let

be the isotropy group of the cusp $k_{i}$. Below we give a more precise description of these isotropy groups. First, the isotropy group of $\infty$ is easy to compute: For $\gamma=\iota\left(\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\right)\in\Gamma_{K}$, by direct computation $\gamma\cdot\iota(\infty)=\iota(\infty)$ if and only if $c=0$, that is

where $B<\mathrm{H}$ is the subgroup of upper triangular matrices in $\mathrm{H}$.

To compute the other isotropy groups, we will translate the cusp $k_{i}$ to $\infty$. For any integral ideal $I\subset\mathcal{O}_{K}$ let

For each $1\leq i\leq\kappa$ since $I_{i}=\langle a_{i},c_{i}\rangle$, we can find $b_{i},d_{i}\in I_{i}^{-1}$ such that $a_{i}d_{i}-b_{i}c_{i}=1$. Throughout the paper we fix the scaling matrix

Note that $\xi_{i}$ satisfies $\xi_{i}\iota(\infty)=\iota(k_{i})$. We then have the following description of $\Gamma_{i}$.