The Minimal Denominator Function and Geometric Generalizations

A main result of [4] is the computation of the asymptotics of the expected value of $q_{\min}(x,\delta)$ when $x$ is chosen randomly uniformly with respect to the Lebesgue measure on $[0,1]$ and $\delta\rightarrow 0$.

$$\mathbb{E}_{x}\left[q_{\min}(x,\delta)\right]=\int_{0}^{1}q_{\min}(x,\delta)dx.$$

(1.3)

Let $X_{m}=SL(m,\mathbb{R})/SL(m,\mathbb{Z})$ denote the space of unimodular lattices in $\mathbb{R}^{m}$ equipped with the measure $\mu_{m}$, arising from the Haar measure on $SL(m,\mathbb{R})$ normalized so that $\mu_{m}$ is a probability measure on $X_{m}$. Define the function $F^{m}_{1}:X_{m+1}\rightarrow\mathbb{R}$ by

$$F^{m}_{1}(\Lambda)=\min\left\{x>0:\text{ there exists }\mathbf{y}\in\mathbb{R}^{m}\text{ such that }\begin{bmatrix}x\\ \mathbf{y}\end{bmatrix}\in\Lambda\cap C_{1}\right\},$$

(1.5)

where $C_{1}=\left\{\begin{bmatrix}x&\mathbf{y}\end{bmatrix}^{T}\in\mathbb{R}^{m}:x>0\text{ and }\mathbf{y}\in\mathbb{R}^{m},\norm{\mathbf{y}}_{m}<x\right\}$ and $\norm{\cdot}_{k}$ denotes the max norm on $\mathbb{R}^{k}$.

A key application of our interpretation of $q_{\min}(x,\delta)$ is the following theorem which describes the limiting distribution of the normalized function $\delta^{\frac{1}{2}}q_{\min}(x,\delta)$ in terms of the function $F^{1}_{1}(\Lambda)$, where $\Lambda$ is a random unimodular lattice chosen from $X_{2}$ according to the probability measure $\mu_{2}$.

The framework developed above can be generalized in two ways. One, motivated by Diophantine approximation, is to consider higher dimensional versions of the minimal denominator function and connections to dynamics on space of unimodular lattices. Another is to consider different discrete subsets of the plane which arise from geometric constructions, in particular, we look at holonomy vectors of saddle connections on translation surfaces. We explore higher-dimensions in §5 and discuss saddle connections in §6.

For $m\in\mathbb{N}$, $\textbf{x}\in[0,1]^{m}$, and $\delta>0$, we define

$$Q^{m}(\textbf{x},\delta)=\min\left\{q\in\mathbb{N}:\text{ there exists }\textbf{p}\in\mathbb{Z}^{m}\text{ such that }\norm{\textbf{x}-\frac{1}{q}\textbf{p}}_{m}<\delta\right\}.$$

$Q^{m}(\mathbf{x},\delta)$ computes the smallest least common multiple of the denominators of the rational points in a $\delta$ neighborhood of the point $x$ in $[0,1]^{m}.$ In particular, $Q^{m}$ measures the complexity of rational points in a $\delta$ neighborhood of each point in the $m$-dimensional unit cube. In this aspect we see that $Q^{1}(x,\delta)=q_{\min}(x,\delta)$. We then have the following generalization of Theorem 1.2

A compact translation surface is an ordered pair $(X,\omega)$ where $X$ is a compact Riemann surface and $\omega$ is a non-zero holomorphic 1-form on $X$. We usually write $\omega$ to represent the translation surface $(X,\omega)$ for convenience and write $(X,\omega)$ if clarification is needed. A geometric way to think about a translation surface is as follows: Let $P=\bigsqcup_{i=1}^{k}P_{i}$ be a disjoint union of connected polygons (not necessarily convex) $P_{i}\subset\mathbb{C}$ such that the collection of the sides can be partitioned into parallel pairs $\{s_{a},s_{b}\}$ of the same length; which are subsequently identified via the Euclidean translation $T_{a,b}$ sending $s_{a}$ onto $s_{b}$, to produce a surface. Since translations are holomorphic, the surface obtained from this procedure can be endowed with a complex structure. Moreover, since for each $c\in\mathbb{C}$ we have that $d(z+c)=dz$, the holomorphic $1$-form $dz$ on the plane descends to this surface and endows it with a (non-zero) holomorphic one form $\omega$.

Let $\left\{z_{1},z_{2},...,z_{j}\right\}$ be the zeros of $\omega$ on $X$. $\omega$ induces a flat Riemannian metric on the surface $X\setminus\left\{z_{1},...,z_{j}\right\}$. A saddle connection is a geodesic $\gamma$ starting and ending at two (possibly the same) zeros of $\omega$ without passing through any other zeros.

We may record saddle connections as complex numbers as follows: given a saddle connection $\gamma$ on a translation surface $\omega$, we define the holonomy vector of $\gamma$ by

$$z_{\gamma}=\int_{\gamma}\omega\in\mathbb{C}.$$

The holonomy vector

$$z_{\gamma}=x_{\gamma}+iy_{\gamma}$$

records the horizontal ($x_{\gamma}$) and vertical ($y_{\gamma}$) displacement of $\gamma$ on $\omega$. Denote the collection of all holonomy vectors of $\omega$ by

$$\Lambda_{\omega}:=\{z_{\gamma}:\gamma\mbox{ is a saddle connection on }\omega\}.$$

Masur showed in [15] that for any $\omega$, there exists positive constants $c_{1}$ and $c_{2}$ (only dependent on $\omega$) such that

$$c_{1}R^{2}\leq\absolutevalue{\Lambda_{\omega}\cap B(0,R)}\leq c_{2}R^{2},$$

characterizing the growth rate of the $\Lambda_{\omega}$. Much work has been done since then to understand the distribution of saddles for different kinds of translation surfaces.

Let $\omega$ be a translation surface and $\delta>0$ and consider the following function:

$$\Psi(\omega,\delta)=\min\left\{\real(z_{\gamma}):\gamma\text{ is a saddle connection of }\omega\text{ and }z_{\gamma}\in\Lambda(\omega)\cap C_{\delta}\right\},$$

where $C_{\delta}=\{x+iy\in\mathbb{C}:x>0\text{ and }y<\delta x\}$.

The next result computes the limiting distribution of $\sqrt{\delta}\Psi(\omega,\delta)$ as $\delta\rightarrow 0$ whenever $\omega$ is a Veech surface with Veech group $\Gamma_{\omega}$. We also set the notation that $Y_{\omega}=SL(2,\mathbb{R})/\Gamma_{\omega}$ and $\mu_{\omega}$ is the induced measure on $Y_{\omega}$ by the Haar measure on $SL(2,\mathbb{R})$. Denote the matrix $\begin{bmatrix}1&0\\ -\alpha&1\end{bmatrix}$ by $h_{\alpha}$. We then have the following result on the distribution of short saddle connections.

In the 1920s Franel [7] and Landau [13] restated the Riemann Hypothesis as a problem on the distribution of the Farey Sequence in $[0,1]$. This contributed to the growth in interest of questions on the distribution of rational numbers with small denominators. For instance, Hall computed the distribution of the spacing between consecutive Farey fractions, when properly normalized [8]. More recently, Boca-Zaharescu computed correlation formulas for the Farey sequence in [3]. Theorem 1.2 describes the distributions of the waiting time to for the Farey Sequence to intersect a randomly chosen small interval in $[0,1]$, expanding on the work of Chen-Haynes in [4].

Given a lattice $\Lambda$ in $\mathbb{R}^{n}$ and a norm, $\norm{\cdot}$, on $\mathbb{R}^{n}$, we may ask the following question: What is the shortest non-zero vector in $\Lambda$? This question is known as the short vector problem (SVP) and it has been of interest in cryptography. This problem is NP-hard. We create a dictionary that allows us to connect the minimal denominator function to the minimization of the lengths of vectors inside a thin cone. Other variants of the SVP have been studied in the past. The work of Siegel [17] allows us to compute the average number of intersections between a randomly chosen unimodular lattice and a region of the $\mathbb{R}^{n}$. More recently, Kim [10] computed the distribution for the lengths of the the first $k$-short vectors in a randomly chosen lattice.

Translation surfaces arise naturally from problems in rational billiards, Riemann surfaces, and number theory. Masur [15] characterized the growth rate of the set of their holonomy vectors in 1990. Over the past few decades a lot of work has been done to get finer statistics on the distribution of saddle connections of different kinds of translation surfaces. Athreya-Chaika [1] described the decay of smallest angle gaps between saddle connections in almost any surface. More recently, Kumanduri-Sanchez-Wang proved the existence of gap distribution for any Veech surface in [12]. A consequence of our setup and generalizations is found in Theorem 1.4, where we compute the limiting distribution of the length of the shortest saddle connection in a thin cone.

I would like to thank my advisor, Jayadev Athreya, for introducing me to this collection of ideas, their exceptional patience, enthusiasm, and unconditional support.

We specialize the setup above by looking at sequences of $G$-equivariant processes over a fixed space and a fixed equivariant process map. Let $(X,\lambda_{n},\nu)$ be such a sequence of $G$-equivariant processes. Then we have the following result which allows us to exchange weak^∗ convergence results for equidistribution results.