# Hamiltonian Dynamics and Structural States in Two Dimensional Active Systems

Yuval Shoham<EMAIL_ADDRESS>School of Physics and Astronomy and
the Center for Physics and Chemistry of Living Systems, Tel Aviv University,
Tel Aviv 6997801, Israel  Naomi Oppenheimer<EMAIL_ADDRESS>School of
Physics and Astronomy and the Center for Physics and Chemistry of Living
Systems, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

We show that a two dimensional system of active particles that interact
hydrodynamically such as microswimmers can be expressed using a unifying
Hamiltonian formalism. We simulate a random arrangement of co-oriented
microswimmers and find they evolve into sharp lines at a particular tilt. We
call these states ”escalators” as particles circulate along these canted
conveyor belts. We argue that the conservation of the Hamiltonian and its
symmetry germinate the self-assembly of the observed steady-state
arrangements. The Hamiltonian is scale-invariant and depends strictly on the
angles between the particles and their swimming orientation, thereby
restricting their available phase-space. Stability analysis confirms that
alignment at either a low or a high angle is unstable, and the ensemble of
swimmers breaks into escalators.

###### pacs:

Out-of-equilibrium systems, be they biological or artificial, are usually
suspended in a fluid. The hydrodynamic interactions between them are long-
ranged, and when confined to a two-dimensional (2D) plane, even extremely
long-ranged. While, in general, microorganisms live in a three-dimensional
environment Saintillan and Shelley (2007); Spagnolie and Lauga (2012); Lauga
_et al._ (2021), many artificial and biological systems have lower effective
dimensions Lushi  _et al._ (2014); Miles  _et al._ (2019). As an example, the
membrane of the cell is a thin fluid film made of lipids Robertson (1981). The
dynamics in the plane of the membrane can be described as an effective 2D
fluid Saffman and Delbruck (1975); Ramachandran  _et al._ (2010); Brown
(2011); Oppenheimer and Diamant (2009). The membrane is, in fact, not a bare,
simple fluid but a very complex fluid — with a high area fraction of proteins.
Some proteins actively move or apply stresses on the membrane, generating
flows that decay as $\sim r^{-1}$ Drescher  _et al._ (2010); Lenz  _et al._
(2003); Oppenheimer  _et al._ (2019); Manikantan (2020). Artificial examples
of active swimmers have been prevalent in recent years, allowing fine-tuning
of the parameter space, and thereby expanding the study of non-equilibrium
phenomena Vutukuri  _et al._ (2020a); Palacci  _et al._ (2014, 3222); Dreyfus
_et al._ (2005); Paxton  _et al._ (2004); Howse  _et al._ (2007); Wang  _et
al._ (2013); Sridhar  _et al._ (2020); Bechinger  _et al._ (2016); Sacanna
_et al._ (2010); Zhang  _et al._ (2009); Zion  _et al._ (2020). A possible
two-dimensional active system could be generated in Giant Unilamellar Vesicles
Vutukuri  _et al._ (2020b), in soap films Vivek and Weeks (2015); Prasad and
Weeks (2008), or in smectic films Nguyen  _et al._ (2010); Radzihovsky  _et
al._ (2017); Eremin  _et al._ (2011).

We show that for active particles in a 2D fluid, the equations of motion give
rise to a geometric Hamiltonian description. We further show that when
particle’s orientations are aligned, such as in a flocking mode, symmetries of
the Hamiltonian give rise to structural conservation laws, which limit the
angular spread of the particles resulting in sharp lines at a given tilt. This
type of Hamiltonian description has been prevalent for vortices in an ideal
fluid Onsager (1949); Newton (2001); Aref and Pomphrey (1982), more recently
also for rotors in a viscous flow Lenz  _et al._ (2003, 2004); Yeo  _et al._
(2015); Lushi and Vlahovska (2015); Oppenheimer  _et al._ (2019, 2021), for
sedimenting disk arrays Chajwa  _et al._ (2020, 2019), and for a swimmer or
two interacting with a flow or an external field Zöttl and Stark (2012); Stark
(2016); Bolitho  _et al._ (2020).

We consider hydrodynamic interactions between microscopic organisms such that
inertia is negligible, and the governing equations are Stoke’s equations. A
driven active particle is force-free. Thus, to a leading order, it will
generate a flow of a force-dipole given by $0=-\nabla
p+\mu\nabla^{2}\mathbf{v}+\mathbf{D}\bm{:}\nabla\delta(\mathbf{r})$, where
$\mu$ is the viscosity, $\mathbf{D}$ is the magnitude of the force dipole, and
$\bm{:}$ is a double dot product. Assuming incompressibility gives also
$\bm{\nabla}\cdot\mathbf{v}=0$. The resulting flow can be decomposed into an
anti-symmetric part, a “rotlet”, and a symmetric part, a “stresslet”. They are
compactly written in polar coordinates as

$\mathbf{v}=\frac{1}{2\pi
r}\left[T\hat{\theta}+S\cos{(2\theta-2\varphi)}\hat{r}\right],$ (1)

where $T$ is the rotlet strength, $S$ the stresslet strength, $\varphi$ the
orientation of the force-dipole with respect to the $\hat{x}$ axis, and ${\bf
r}=(x,y)=r(cos\theta,sin\theta)$ is the position of the particle. A general
active particle will have a stresslet part, whether it is moving or statically
applying active forces. Previous work focused on the rotational part Lenz  _et
al._ (2003); Lushi  _et al._ (2014); Lushi and Vlahovska (2015); Oppenheimer
_et al._ (2019, 2021). Here we focus on the stresslet part of the flow-field.
The incompressibility equation implies the existence of a vector potential
such that $\mathbf{v}=\bm{\nabla}^{\perp}\psi$, where $\psi$ is the
streamfunction, and
$\bm{\nabla}^{\perp}\equiv\left(\partial_{y},-\partial_{x}\right)$. These
equations of motion are Hamilton equations, with $x$ and $y$ being the
conjugate variables. The streamfunction of the symmetric part of Eq. 1 is
$\psi_{s}=S\sin(2\theta-2\varphi)/\pi$ (see Fig. 1).

In a system of many similar active particles, where each particle is affected
only by the other particle’s flow-field, this streamfunction can be summed to
become the Hamiltonian of the system. The flow-field of the $i^{th}$ particle
is given by,

$\displaystyle\mathbf{v}_{i}=\sum_{j\neq
i}\frac{S_{j}}{2\pi}\frac{\left(x_{i}-x_{j}\right)^{2}-\left(y_{i}-y_{j}\right)^{2}}{r_{ij}^{4}}\mathbf{r}_{ij},$
(2)

where $\mathbf{r}_{ij}\equiv\mathbf{r}_{i}-\mathbf{r}_{j}$ is the vector
pointing from particle $j$ to particle $i$, and $S_{i}$ is the strength of the
$i^{th}$ particle. These velocities can be derived from a Hamiltonian,
$S_{i}\mathbf{v}_{i}=\nabla_{i}H$, where $H$ is

$\displaystyle H=\sum_{\begin{subarray}{c}i,j,i\neq
j\end{subarray}}\frac{S_{i}S_{j}}{2\pi}\sin{2\theta_{ij}},$ (3)

and $\theta_{ij}$ is the relative angle between the vector connecting
stresslets $i$ and $j$ and the $x$-axis (see Fig. 1B). We note four points
about this Hamiltonian: (a) It does not depend on time, therefore, from
Noether’s theorem Noether (1918), it is conserved. Collisions between
particles perturbs this conservation (See Fig. 1C). (b) The Hamiltonian is
scale-invariant, that is, it is the same whether the stresslets are very close
or very far, as long as the relative angles between them are the same (see
Fig. 1B). (c) It is symmetric with respect to $\pm\pi/4$. Therefore, we expect
the solution to be symmetric around that angle. (d) It is symmetric to
translations, therefore ${\bf d}_{\rm act}\equiv\sum_{i}S_{i}{{\bf
r}_{i}}/N={\rm const}$, where ${\bf d}_{\rm act}$ is the “center of activity”
in analogy to the center of mass. In what follows, we consider only stresslets
with the same activity strength $S_{i}=S$.

Figure 1: (a) Streamlines of a single particle swimming along the $x$
direction. Note the stagnation lines at $\pm\pi/4$, the inward flow along the
vertical and outward along the horizontal. (b) Schematics showing six oriented
swimmers, swimming in the $\hat{x}$ direction with the same Hamiltonian
whether at close proximity or far apart as long as the angle between each two
swimmers and their swimming direction, $\theta_{ij}$ is fixed. (c) Hamiltonian
as a function of time for 4 random particles. The Hamiltonian is conserved
between collisions. A collision event is marked in red in the middle panel.
Figure 2: Snapshots from a molecular dynamics simulations of 300 active
particles. Particles are initiated randomly in a square of size $10\times 10$.
Swimming directions are fixed along the $x$ axis. (a) Snapshots from the early
times where the ensemble spreads and elongates. $y$ positions are shifted down
between times for clarity. (b) Snapshots from later times where an instability
is formed and grows. The particles develop sharp lines at $\pm\pi/4$, which we
call escalators as particles circulate around them. Inset shows a zoom in on
one of escalators with overlayed snapshots at different times. particles going
left (right) are marked in blue (green).

A single stresslet does not move by the flow it creates. The simplest dynamic
system is therefore composed of two particles, in which case, the Hamiltonian
is reduced to $H=S^{2}\sin{2\theta}/(2\pi)$, and the relative angle itself is
conserved. The dynamics of two stresslets are confined to the line connecting
them. When placed at initial distance $d_{0}$ and angle $\varphi$ relative to
the $x$-axis, the relative distance between the two particles is
$R=\sqrt{2S\cos{\left(2\varphi\right)}t/\pi+d_{0}^{2}}$. The relative angle
$\varphi$ determines if the two stresslets collide or disperse. For
$\varphi<\pi/4$, the stresslets repel each other. The repulsion scales with
time as $R^{2}\sim t$, similar to diffusion. On the other hand, for
$\varphi>\pi/4$, the stresslets collide after a finite time
$t=d_{0}^{2}\pi/(2S\left|\cos{\left(2\varphi\right)}\right|)$. At exactly
$\varphi~{}=~{}\pi/4$, particles remain static, and the initial distance is
fixed.

A system of many stresslets can no longer be solved analytically. However, the
conservation laws still apply. We numerically integrated Eq. 2 using the
python library `scipy.integrate.DOP853`, which is an $8^{th}$ order Runge-
Kutta method with an adaptive time stepper. As we saw for two swimmers, the
interaction between each two particles is radial and their interaction decays
as $\sim r^{-1}$. They repel or attract depending on their relative angle.
Thus, when two swimmers attract, they accelerate toward each other and
eventually collide, such that the velocity diverges. Actual active particles
have a given size and cannot overlap. We, therefore, introduce soft steric
repulsion of the form $\Delta\mathbf{v}_{steric}=\Delta
t\,k_{steric}\left(l_{steric}-\left|\mathbf{r}\right|\right)\hat{r}$ if
$\left|\mathbf{r}\right|<l_{steric}$ and zero otherwise. When two stresslets
get closer than a certain steric length $l_{steric}$, a repelling force
proportional to their relative distance $\left|\mathbf{r}\right|$ by a very
large spring constant $k_{steric}$ is applied and pushes them apart. In our
simulations, $l_{steric}=0.001$ and $k_{steric}=1,000$. This interaction is
added to the regular stresslet interaction. Due to the steric interactions,
the Hamiltonian is no longer strictly conserved. To ensure that collisions do
not dominate the behavior of the system, we work in the dilute limit, with
average distances between particles much larger than the steric length, where
collisions are rare. We verified that the Hamiltonian is accurately conserved
in between collisions (see Fig. 1C).

We initialize 300 oriented stresslets randomly positioned in a square and let
them evolve, all swimming in the $\hat{x}$ direction. At first, the system
follows the general shape of a single stresslet shown in Fig. 1A— the system
compresses in the $y$-direction and expands in the $x$-direction, into a
“street” of stresslets, see Fig. 2A. At intermediate times, this street shows
instabilities, which grow over time. At long times, these instabilities create
stable shapes where particles concentrate along inclined streets at angles
$\pm\varphi=\pi/4$ (see Fig. 2B), which we term “escalators”. There is
circulation around these escalators, with particles above and below going in
opposite directions. See inset in Fig. 2B, for overlayed snapshots at
preceding times where particles are color coded according to their direction
of motion.

Why are there always escalators of $\varphi=\pm\pi/4$ when the system is
initialized as a random square? We explain this using symmetry and Hamiltonian
conservation arguments. We begin by showing that the system can only evolve
into several escalators and not a single one. The Hamiltonian of the initial
state is, on average, $0$ since the initial angles are random and $H$ can be
written as $H=N(N-1)S^{2}\left<\sin{2\theta_{ij}}\right>/2\pi=0$, where
$\left<.\right>$ is the average over the ensemble and $N$ is the number of
stresslets. The Hamiltonian of an escalator with $\varphi=\pm\pi/4$, on the
other hand, is non-zero. In fact it has maximal magnitude. Given that the
Hamiltonian is conserved, a system of stresslets scattered in a square cannot
develop into a single escalator. Indeed, we notice that the system did not
decompose into a single escalator, but many escalators, each with an
inclination angle $\varphi=\pm\pi/4$. A simple case for this decomposition is
two escalators with opposing inclination angles $\varphi=\pm\pi/4$, placed at
opposing sides of the $y$ axis in symmetric form, see Fig. 1B. Mirroring the
system along the $y$-axis, the Hamiltonian is
$H^{\text{mirror}_{y}}\propto\left<\sin{\left(2\cdot\left(\pi-\theta_{ij}\right)\right)}\right>=-H$,
which implies again that the Hamiltonian is zero. Thus, a symmetric
combination of stresslets conserves the Hamiltonian. Moreover, there is no
limitation to the number of escalators the system can develop into, and as
seen in Fig. 2, the system indeed develops into many escalators of opposing
inclination angles. The Hamiltonian is symmetric around $\pm\pi/4$, so the
steady state configuration needs to exhibit this symmetry. We go on to test
the stability of particles aligned at different angles.

Stability Analysis For a Street of stresslets.

Figure 3: Stability of particles at different angles under perturbation. We
initiate 200 particles randomly on a line tilted at three different angles:
$\pi/8$ in green, $\pi/4$ in black and $\pi/3$ in green. We add a small sine-
wave perturbation to their positions and let the system evolve over time. (a)
Initial positions, (b) Configuration after a short period ($t=0.1$) shows all
initial conditions resulted in escelators at about $\pi/4$ degrees. Note how
the $\pi/8$ street breaks into smaller streets. (c) Result at long times with
all initial conditions resulting in lines at $\sim 40^{\circ}$. Slightly less
than $\pi/4$ and consistent with around $15\%$ decrease in the value of the
Hamitlonian. At later times the system continues to spread but maintains these
angles as there are hardly any collisions. (d) The relative error of the
Hamitlonian as a function of time, where $H_{0}$ is its initial value.

We use three different methods to test the stability of an escalator, inclined
by an angle $\varphi$. First, we use linear stability analysis and show that
the stability of the inclined escalator is of non-linear nature, as the first
order of perturbation gives a fixed point of type center, meaning that all
escalators are neutrally stable up to first order. Next, we use numeric
simulations of escalators with different inclination angles to show that
escalators are unstable — except for $\varphi=\pi/4$ inclination. Lastly, by
calculating analytically the velocity field created by a continuous
distribution of particles in an escalator, we show that for $\varphi<\pi/4$,
there is a repelling force pushing particles away from it. In addition,
escalators at $\varphi>\pi/4$ have an attractive force that makes them
collapse and are therefore, inherently unstable. We combine the results to
conclude that the only stable escalator is $\varphi=\pm\pi/4$.

We follow the method of linear stability analysis used in Ref. Saffman (1993)
§8. We start with a continuous line of stresslets with strength density $S$.
In complex notation the velocity is

$\displaystyle\dot{Z}=\frac{S}{2\pi}\frac{1}{2Z}\left(1+\left(\frac{Z}{\bar{Z}}\right)^{2}\right),$
(4)

where $Z=x+iy$, and $\bar{Z}=x-iy$. Along the parametric line
$\left\\{Z_{s}=Z\left(s\right)|s\in\left(-\infty,\infty\right)\right\\}$, the
velocity at each point $\dot{Z}_{s}$ is

$\displaystyle\dot{Z}_{s}=\int_{-\infty}^{\infty}\frac{S}{4\pi\left(Z_{s}-Z_{s^{\prime}}\right)}\left(1+\left(\frac{Z_{s}-Z_{s^{\prime}}}{\bar{Z}_{s}-\bar{Z}_{s^{\prime}}}\right)^{2}\right)\mathrm{d}s^{\prime}.$
(5)

For a stresslet “street” with an inclination $\varphi$ compared to the
$x$-axis, $Z\left(s\right)=se^{i\varphi}$. Adding a small Fourier-decomposed
perturbation $\varepsilon_{s}=\sum_{q}a_{q}e^{iqs}$ to the street, keeping
only its lowest non-zero order, and equating orthogonal phasors, the equations
of motion for $Z$ and $\bar{Z}$ (Eq. 5) become

$\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}a_{q}\\\
\bar{a}_{-q}\end{pmatrix}=-\frac{Sq}{2}\begin{pmatrix}i\sin{4\varphi}&-e^{4i\varphi}\\\
e^{-4i\varphi}&i\sin{4\varphi}\end{pmatrix}\begin{pmatrix}a_{q}\\\
\bar{a}_{-q}\end{pmatrix},$ (6)

with eigenvalues $\lambda_{1,2}=-\frac{1}{2}iSq\left(\sin{4\varphi}\pm
1\right)$. Note that the eigenvalues are imaginary and create a neutrally
stable point of type center. Therefore, in the linear limit, small
perturbations will not grow nor decay. A similar calculation in a 3D fluid
showed that stresslets in that case are linearly unstable Lauga  _et al._
(2021).

Next, we use numeric simulations of perturbed escalators. We initialize 200
stresslets in random locations along tilted lines at a fixed angle and perturb
them by adding a sine wave to their locations in the perpendicular direction.
We then let the system develop over time. After a short time ($t=0.1$),
escalators with an inclination angle $\varphi\neq\pi/4$ are broken and end up
as a system of many escalators approaching $\varphi=\pi/4$ (Fig. 3). The
result at long times ($t\sim 100$, Fig. 3) is, though, not exactly $\pi/4$,
due to a decrease in the Hamiltonian (see Fig. 3D). Such a decrease in the
Hamiltonian comes from the steric interactions introduced to the system (Fig.
1C), which are necessary to avoid the divergence of the velocity field at
short distances since $v\sim 1/r$. From its initial value, the Hamiltonian
loses $\sim 15\%$ at $t=100$. This change seems to scale sub-linearly with
time, i.e. $\leq\sqrt{t}$ (see Supplementary Information). In a way, the
system diffuses into another state due to collisions. Let us attempt to take
into account the decrease in the Hamiltonian and estimate the angle of a
“perfect” escalator with the same decreased Hamiltonian. The resulting angle
is aroung $40^{\circ}$ and agrees well with the simulation predictions, as
shown in the red dashed line in Fig. 3.

Contrary to the result obtained by linear analysis, the only stable escalator
is the one with an inclination angle of about $\varphi=\pi/4$. When a street
of any inclination is infinite, and the stresslets are at equal distances
apart, the system is completely stationary. On the other hand, when a street
is finite, only an escalator with inclination $\pm\pi/4$ will remain
stationary, because of the stagnation lines at $\theta=\pm\pi/4$. This already
hints that an unperturbed escalator of inclination $\pm\pi/4$ is more stable
than any other unperturbed street. To test analytically the stability, we use
a continuous model of an infinite escalator where particles are equispaced and
ask what flow field it creates — if its flow field repels particles away from
it, it is certainly not stable. We sum an infinite series of stresslets laid
on a line of inclination $\varphi$ distanced $L$ apart. We calculate the
velocity that the system of stresslets creates at the time of initiation.
Instead of inclining the street, we look at stresslets on the $x$-axis whose
direction is rotated by $\varphi$. The velocity due to a single stresslet is
given by the radial part of Eq. 1. The velocity at a distance $h$ above the
escalator is given by an infinite sum over all stresslets which gives ${\bf
v}=\left[\left(\coth{\rho}-\rho\csch^{2}{\rho}\right)\sin{2\varphi},\rho\csch^{2}{\rho}\cos{2\varphi}\right]S/2L$
with $\rho=\pi h/L$. The $y$-component represents the instability of the line.
We see there will be a repelling force from the inclined street when
$\cos{2\varphi}>0$. This means that for $\cos{2\varphi}>0$ (i.e.
$\varphi<\pi/4$), the inclined street is unstable to perturbations. For
$\cos{2\varphi}=0$, that is $\varphi=\pi/4$, the velocity outside the
escalator is zero, therefore a perturbation neither grows nor decays. What
happens for angles larger than $\pi/4$? In that case, as we have seen for two
stressletss, there is an attraction between the particles. Because there are
no repelling forces other than collisions, a stresslet street at such an angle
collapses onto itself and loses its stability. We finally conclude that indeed
$\varphi=\pi/4$ is the only stable angle.

Discussion. In this work, we introduced a new method to describe a confined
system of microswimmers robustly, using multipole expansion and a Hamiltonian
formalism, which can be used to unveil the dynamics of many microbiological
and other synthetic active systems. The symmetries of the Hamiltonian limit
the possible steady state configurations of the the particles. Numeric
simulations of systems of many stresslets gave further insight into the
dynamics of such systems. By simulating identical stresslets in a random
square as an initial condition, we found that the system first flattens into a
street but then forms instabilities, which evolve into more stable shapes
—“escalators” in which active particles move in a conveyor-belt-like motion
along an inclined line. Using the conservation of the Hamitonian and several
stability analysis methods, we showed that only $\pm\frac{\pi}{4}$ inclination
angles of escalators were stable and could be formed spontaneously from the
initial condition. Future directions of this work include looking into
populations of stresslets — a stresslet is determined by two parameters, its
strength $S$ and its rotation $\theta$. The dynamics can differ drastically by
dividing systems into populations of different parameters.

Acknowledgments. We thank Haim Diamant for fruitful discussions. This research
was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1752/20).

## References

  * Saintillan and Shelley (2007) D. Saintillan and M. J. Shelley, Physical review letters 99, 058102 (2007).
  * Spagnolie and Lauga (2012) S. E. Spagnolie and E. Lauga, Journal of Fluid Mechanics 700, 105 (2012).
  * Lauga  _et al._ (2021) E. Lauga, T. N. Dang, and T. Ishikawa, Europhysics Letters 133, 44002 (2021).
  * Lushi  _et al._ (2014) E. Lushi, H. Wioland, and R. E. Goldstein, Proceedings of the National Academy of Sciences 111, 9733 (2014).
  * Miles  _et al._ (2019) C. J. Miles, A. A. Evans, M. J. Shelley, and S. E. Spagnolie, Physical Review Letters 122, 098002 (2019).
  * Robertson (1981) J. D. Robertson, Journal of Cell Biology 91, 189s (1981).
  * Saffman and Delbruck (1975) P. G. Saffman and M. Delbruck, Proceedings of the National Academy of Sciences 72, 3111 (1975).
  * Ramachandran  _et al._ (2010) S. Ramachandran, S. Komura, M. Imai, and K. Seki, The European Physical Journal E 31, 303 (2010).
  * Brown (2011) F. L. Brown, Quarterly Reviews of Biophysics 44, 391 (2011).
  * Oppenheimer and Diamant (2009) N. Oppenheimer and H. Diamant, Biophysical Journal 96, 3041 (2009).
  * Drescher  _et al._ (2010) K. Drescher, R. E. Goldstein, N. Michel, M. Polin, and I. Tuval, Physical Review Letters 105, 168101 (2010).
  * Lenz  _et al._ (2003) P. Lenz, J.-F. Joanny, F. Jülicher, and J. Prost, Physical Review Letters 91, 108104 (2003).
  * Oppenheimer  _et al._ (2019) N. Oppenheimer, D. B. Stein, and M. J. Shelley, Physical Review Letters 123, 148101 (2019).
  * Manikantan (2020) H. Manikantan, Physical review letters 125, 268101 (2020).
  * Vutukuri  _et al._ (2020a) H. R. Vutukuri, M. Lisicki, E. Lauga, and J. Vermant, Nature Communications 11, 2628 (2020a).
  * Palacci  _et al._ (2014) J. Palacci, S. Sacanna, S.-H. Kim, G.-R. Yi, D. J. Pine, and P. M. Chaikin, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, 20130372 (2014).
  * Palacci  _et al._ (3222) J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Science. 339, 936 (2013222).
  * Dreyfus  _et al._ (2005) R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J. Bibette, Nature 437, 862 (2005).
  * Paxton  _et al._ (2004) W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. S. Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert, and V. H. Crespi, Journal of the American Chemical Society 126, 13424 (2004).
  * Howse  _et al._ (2007) J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Physical Review Letters 99, 048102 (2007).
  * Wang  _et al._ (2013) W. Wang, W. Duan, S. Ahmed, T. E. Mallouk, and A. Sen, Nano Today 8, 531 (2013).
  * Sridhar  _et al._ (2020) V. Sridhar, F. Podjaski, J. Kröger, A. Jiménez-Solano, B.-W. Park, B. V. Lotsch, and M. Sitti, Proceedings of the National Academy of Sciences 117, 24748 (2020).
  * Bechinger  _et al._ (2016) C. Bechinger, R. D. Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Reviews of Modern Physics 88, 045006 (2016).
  * Sacanna  _et al._ (2010) S. Sacanna, W. T. M. Irvine, P. M. Chaikin, and D. J. Pine, Nature 464, 575 (2010).
  * Zhang  _et al._ (2009) L. Zhang, J. J. Abbott, L. Dong, B. E. Kratochvil, D. Bell, and B. J. Nelson, Applied Physics Letters 94, 064107 (2009).
  * Zion  _et al._ (2020) M. Y. B. Zion, A. Modin, Y. Caba, and P. M. Chaikin, (2020).
  * Vutukuri  _et al._ (2020b) H. R. Vutukuri, M. Hoore, C. Abaurrea-Velasco, L. van Buren, A. Dutto, T. Auth, D. A. Fedosov, G. Gompper, and J. Vermant, Nature 586, 52 (2020b).
  * Vivek and Weeks (2015) S. Vivek and E. R. Weeks, PLOS ONE 10, e0121981 (2015).
  * Prasad and Weeks (2008) V. Prasad and E. R. Weeks, (2008).
  * Nguyen  _et al._ (2010) Z. H. Nguyen, M. Atkinson, C. S. Park, J. Maclennan, M. Glaser, and N. Clark, Physical Review Letters 105, 268304 (2010).
  * Radzihovsky  _et al._ (2017) S. P. Radzihovsky, C. Cranfill, Z. Nguyen, C. S. Park, J. E. Maclennan, M. A. Glaser, and N. A. Clark, Soft Matter 13, 6314 (2017).
  * Eremin  _et al._ (2011) A. Eremin, S. Baumgarten, K. Harth, R. Stannarius, Z. H. Nguyen, A. Goldfain, C. S. Park, J. E. Maclennan, M. A. Glaser, and N. A. Clark, Physical Review Letters 107, 268301 (2011).
  * Onsager (1949) L. Onsager, Il Nuovo Cimento 6, 279 (1949).
  * Newton (2001) P. K. Newton,  _The N-Vortex Problem_, Vol. 145 (Springer New York, 2001).
  * Aref and Pomphrey (1982) H. Aref and N. Pomphrey, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 380, 359 (1982).
  * Lenz  _et al._ (2004) P. Lenz, J.-F. Joanny, F. Jülicher, and J. Prost, The European Physical Journal E 13, 379 (2004).
  * Yeo  _et al._ (2015) K. Yeo, E. Lushi, and P. M. Vlahovska, Physical review letters 114, 188301 (2015).
  * Lushi and Vlahovska (2015) E. Lushi and P. M. Vlahovska, Journal of Nonlinear Science 25, 1111 (2015).
  * Oppenheimer  _et al._ (2021) N. Oppenheimer, D. B. Stein, M. Y. B. Zion, and M. J. Shelley,  (2021).
  * Chajwa  _et al._ (2020) R. Chajwa, N. Menon, S. Ramaswamy, and R. Govindarajan, Physical Review X 10, 041016 (2020).
  * Chajwa  _et al._ (2019) R. Chajwa, N. Menon, and S. Ramaswamy, Physical Review Letters 122, 224501 (2019).
  * Zöttl and Stark (2012) A. Zöttl and H. Stark, Physical Review Letters 108, 218104 (2012).
  * Stark (2016) H. Stark, The European Physical Journal Special Topics 225, 2369 (2016).
  * Bolitho  _et al._ (2020) A. Bolitho, R. Singh, and R. Adhikari, Physical Review Letters 124, 088003 (2020).
  * Noether (1918) E. Noether, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1918, 235 (1918).
  * Saffman (1993) P. G. Saffman,  _Vortex Dynamics_ (Cambridge University Press, 1993).


# Square-roots and lattices

Jens Marklof  Jens Marklof, School of Mathematics, University of Bristol,
Bristol BS8 1UG, U.K.
<EMAIL_ADDRESS>
(Date: 13 June 2024)

###### Abstract.

We construct a point set in the Euclidean plane that elucidates the
relationship between the fine-scale statistics of the fractional parts of
$\sqrt{n}$ and directional statistics for a shifted lattice. We show that the
randomly rotated, and then stretched, point set converges in distribution to a
lattice-like random point process. This follows closely the arguments in
Elkies and McMullen’s original analysis for the gap statistics of
$\sqrt{n}\bmod 1$ in terms of random affine lattices [Duke Math. J. 123
(2004), 95–139]. There is, however, a curious subtlety: the limit process
emerging in our construction is not invariant under the standard
$\operatorname{SL}(2,{\mathbb{R}})$-action on ${\mathbb{R}}^{2}$.

Research supported by EPSRC grant EP/W007010/1. Data supporting this study are
included within the article. MSC2020: 11K06,37D40,60G55

##  1\. Introduction

Square-roots: In their landmark paper [4], published twenty years ago, Elkies
and McMullen proved that the gap distribution for the fractional parts of
$\sqrt{n}$ ($n=1,\ldots,N$, $N\to\infty$) converges to the previously unknown
limiting distribution plotted as the continuous curve in Figure 1 (left). To
state their result more precisely, let us denote by
$0\leq\xi_{1}\leq\xi_{2}\leq\ldots\leq\xi_{N}<1$ the fractional parts of
$\sqrt{n}$ ($n=1,\ldots,N$) ordered by size, and furthermore let
$s_{n}=\xi_{n+1}-\xi_{n}$ be the gap between $\xi_{n}$ and $\xi_{n+1}$
($n=1,\ldots,N-1$) and $s_{N}=1+\xi_{1}-\xi_{N}$ the gap between $\xi_{N}$ and
$1+\xi_{1}$ (think of the unit interval $[0,1)$ as the real line mod 1 with
the end points $0$ and $1$ identified). It is important to note that $\xi_{n}$
and $s_{n}$ will change as we move to a different $N$. The Elkies-McMullen
theorem then states that, for every $s>0$,

(1) $\lim_{N\to\infty}\frac{\\#\\{n\leq N\mid
Ns_{n}>s\\}}{N}=\int_{s}^{\infty}P(s)\,ds,$

where the gap density $P(s)$ is a piecewise analytic function with a power-law
tail. The scaling of $s_{n}$ by $N$ is necessary in view of the average gap
size of $1/N$. It is remarkable that if we change the square-root to any other
fractional power $n^{\beta}$ with $0<\beta<1$, the gap distribution of the
fractional parts appears to have instead an exponential limit density
$P(s)=\mathrm{e}^{-s}$ – the same as for a Poisson point process! This
observation is purely conjectural, and mainly based on numerical experiments.
The best rigorous results in this direction are currently due to Lutsko,
Sourmelidis and Technau [8], who showed that the two-point correlation
function (a slightly weaker fine-scale statistics) of $n^{\beta}$ converges to
that of a Poisson point process, when $0<\beta\leq\frac{1}{3}$. We refer the
reader to [8, 22] for further background and references on the pseudo-random
properties of related arithmetic sequences.

Figure 1. The distribution of gaps in the sequence $\sqrt{n}\bmod 1$,
$n=1,\ldots,7765$ (left) and in the directions of the vectors
$(m-\sqrt{2},n)\in{\mathbb{R}}^{2}$ with $m\in{\mathbb{Z}}$,
$n\in{\mathbb{Z}}_{\geq 0}$, $(m-\sqrt{2})^{2}+n^{2}<4900$ (right). The
continuous curve is the Elkies-McMullen distribution. Reproduced from [11].

Lattices: Let us now consider a given Euclidean lattice
${\mathcal{L}}\subset{\mathbb{R}}^{2}$ of full rank; the simplest example to
keep in mind is ${\mathcal{L}}={\mathbb{Z}}^{2}$. We are interested in the
directional statistics of the lattice points of ${\mathcal{L}}$ as viewed from
a fixed observer located at ${\text{\boldmath$q$}}\in{\mathbb{R}}^{2}$. Let
${\text{\boldmath$y$}}_{1},\ldots,{\text{\boldmath$y$}}_{N}$ denote the first
$N$ shortest vectors in the shifted lattice
${\mathcal{L}}-{\text{\boldmath$q$}}$ (if there are two or more vectors of the
same length, pick your favourite order), and record their angles (relative to
the horizontal axis, say) as $\theta_{1},\ldots,\theta_{N}\in[0,2\pi)$.
Dividing by $2\pi$ and ordering by size produces a set of $\xi_{n}$ in
$[0,1)$, as above for the fractional parts. One of the findings of [11] is
that (a) the gap distribution also converges for this new sequence, and (b)
its limit density agrees with the Elkies-McMullen distribution if and only if
${\text{\boldmath$q$}}\notin{\mathbb{Q}}{\mathcal{L}}$. This is illustrated in
Figure 1 (right) for ${\mathcal{L}}={\mathbb{Z}}^{2}$ and
${\text{\boldmath$q$}}=(\sqrt{2},0)$.

The fundamental reason why the two limit distributions are the same is that
they follow from the equidistribution of two different unipotent translates on
the space of affine lattices which both converge to the same invariant
measure, and are integrated against the same test function. In the present
paper we will provide a more intuitive explanation of this surprising
phenomenon by formulating the Elkies-McMullen convergence in terms of a
natural point process in ${\mathbb{R}}^{2}$ (which is different from the one
considered in their original paper). The idea is to construct a point set in
${\mathbb{R}}^{2}$ such that (a) its directions exactly reproduce the
fractional parts of $\sqrt{n}$ and (b) it is approximated locally by affine
lattices.

For other aspects of the statistics of $\sqrt{n}\bmod 1$ we refer the reader
to [2, 5, 9, 17, 19]; for the distribution of directions in affine lattices,
see [3, 7, 11, 21]; for general background and applications of spherical
averages of point sets, see the introduction of [14].

##  2\. Random point sets

In the following, elements of ${\mathbb{R}}^{2}$ are represented as row
vectors. Define the $2\times 2$ matrices

(2) $D(T)=\begin{pmatrix}T^{-1/2}&0\\\ 0&T^{1/2}\end{pmatrix},\qquad
k(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix},$

and let $\theta$ be a random variable on ${\mathbb{R}}/2\pi{\mathbb{Z}}$
distributed according to an absolutely continuous probability measure
$\lambda$. Note that $D(T)$ and $k(\theta)$ have unit determinant and are thus
elements of the special linear group $\operatorname{SL}(2,{\mathbb{R}})$. For
fixed ${\mathcal{L}}$, $q$ as above, and $T>0$, the random set
$\Xi_{T}=({\mathcal{L}}-{\text{\boldmath$q$}})k(\theta)D(T)$ defines a point
process in ${\mathbb{R}}^{2}$, i.e., a random counting measure that assigns a
unit mass to the location of every element in $\Xi_{T}$. By abuse of notation,
we will use the same symbol for a random point set and the corresponding point
process. For more background on point processes and random point sets, see
[20, Section 4]. A key observation of [11] is that the convergence of the gap
distribution for the directions follows from the convergence (in distribution)
of the point processes $\Xi_{T}\Rightarrow\Xi$ for $T\to\infty$, where $\Xi$
is distributed according to the unique
$\operatorname{ASL}(2,{\mathbb{R}})$-invariant probability measure on the
space of affine lattices. Here $\operatorname{ASL}(2,{\mathbb{R}})$ denotes
the semidirect product group
$\operatorname{SL}(2,{\mathbb{R}})\ltimes{\mathbb{R}}^{2}$ with multiplication
law
$(M,{\text{\boldmath$\xi$}})(M^{\prime},{\text{\boldmath$\xi$}}^{\prime})=(MM^{\prime},{\text{\boldmath$\xi$}}M^{\prime}+{\text{\boldmath$\xi$}}^{\prime})$.
The (right) action of $(M,{\text{\boldmath$\xi$}})$ on ${\mathbb{R}}^{2}$ is
defined by
${\text{\boldmath$x$}}\mapsto{\text{\boldmath$x$}}M+{\text{\boldmath$\xi$}}$,
and the space of affine lattices can be identified with the homogeneous space
$X=\Gamma\backslash G$ with $G=\operatorname{ASL}(2,{\mathbb{R}})$,
$\Gamma=\operatorname{ASL}(2,{\mathbb{Z}})$. We embed
$\operatorname{SL}(2,{\mathbb{R}})$ in $\operatorname{ASL}(2,{\mathbb{R}})$ by
$M\mapsto(M,{\text{\boldmath$0$}})$ and will denote the image also by
$\operatorname{SL}(2,{\mathbb{R}})$.

Figure 2. Left: The point set ${\mathcal{P}}$. Right: A realization of
$\Theta_{T}$ with $T=4$ and $\theta=0.7$.

Figure 3. Left: A realization of the lattice $\Xi_{T}$ in (43) corresponding
to $T=4$ and $\xi=-\frac{\theta}{2\pi}$ with $\theta=0.7$. Right: $\Xi_{T}$
with $\Theta_{T}$ superimposed. This illustrates the approximation of
$\Theta_{T}$ by an affine lattice in fixed bounded subsets of the right half
plane.

Figure 4. Left: A realization of the lattice $\widetilde{\Xi}_{T}$ in (51)
corresponding to $T=4$ and $\xi=\zeta+\frac{1}{2}=-\frac{\theta}{2\pi}$ with
$\theta=0.7$. Right: $\widetilde{\Xi}_{T}$ with $\Theta_{T}$ superimposed.
This illustrates the approximation of $\Theta_{T}$ by an affine lattice in
fixed bounded subsets of the left half plane.

Let us turn to $\sqrt{n}\bmod 1$. Consider the point set

(3)
${\mathcal{P}}=\bigg{\\{}\bigg{(}\sqrt{\frac{n}{\pi}}\cos\big{(}2\pi\sqrt{n}\big{)},\sqrt{\frac{n}{\pi}}\sin\big{(}2\pi\sqrt{n}\big{)}\bigg{)}\,\bigg{|}\,n\in{\mathbb{N}}\bigg{\\}},$

see Figure 2 (left), and the corresponding rotated (through an angle $\theta$)
and stretched (by the linear map $D(T)$) point set

(4) ${\mathcal{P}}k(\theta)D(T)=\bigg{\\{}\bigg{(}\sqrt{\frac{n}{\pi
T}}\cos(2\pi\sqrt{n}-\theta),\sqrt{\frac{Tn}{\pi}}\sin(2\pi\sqrt{n}-\theta)\bigg{)}\,\bigg{|}\,n\in{\mathbb{N}}\bigg{\\}},$

see Figure 2 (right). For $\theta$ random, we denote by
$\Theta_{T}={\mathcal{P}}k(\theta)D(T)$ the corresponding random point set.
Note that ${\mathcal{P}}$ is a Delone set [10] with uniform density in
${\mathbb{R}}^{2}$. That is, for any bounded
${\mathcal{A}}\subset{\mathbb{R}}^{2}$ with boundary of Lebesgue measure zero,

(5) $\lim_{T\to\infty}\frac{\\#({\mathcal{P}}\cap
T{\mathcal{A}})}{T^{2}}=\operatorname{vol}{\mathcal{A}}.$

This follows from the fact that $(\sqrt{n})_{n\in{\mathbb{N}}}$ is uniformly
distributed modulo one, since we can approximate ${\mathcal{A}}$ by finite
unions and intersections of sectors of varying radii; see [10] for details.

Let ${\mathbb{H}}_{-}={\mathbb{R}}_{\leq 0}\times{\mathbb{R}}$ and
${\mathbb{H}}_{+}={\mathbb{R}}_{\geq 0}\times{\mathbb{R}}$ denote the left and
right half plane, respectively. We define the congruence subgroup

(6)
$\Gamma_{2,0}(4)=\bigg{\\{}M\in\operatorname{SL}(2,{\mathbb{Z}})\,\bigg{|}\,M\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\text{or}\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}\bmod
4\bigg{\\}},$

and furthermore the random point set

(7)
$\Theta=\big{(}{\mathbb{Z}}^{2}g\cap{\mathbb{H}}_{+}\big{)}\cup\big{(}-\big{(}\big{[}{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})\big{]}g\big{)}\cap{\mathbb{H}}_{-}\big{)},$

where $g$ is distributed according to the unique $G$-invariant probability
measure $\mu$ on the homogeneous space $Y=\Lambda\backslash G$ with
$\Lambda=\Gamma_{2,0}(4)\ltimes{\mathbb{Z}}^{2}$. Note that

(8)
$(\tfrac{1}{2},-\tfrac{1}{4})\Gamma_{2,0}(4)\in(\tfrac{1}{2},-\tfrac{1}{4})+{\mathbb{Z}}^{2},$

and hence $\big{[}{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})\big{]}g$ is
independent of the choice of representative of $\Lambda g$ in $Y$.

The purpose of this note is to prove the weak convergence
$\Theta_{T}\Rightarrow\Theta$, and to investigate the properties of the limit
process $\Theta$.

######  Theorem 1.

If $\theta$ is random according to an absolutely continuous probability
measure on ${\mathbb{R}}/2\pi{\mathbb{Z}}$, then $\Theta_{T}\Rightarrow\Theta$
as $T\to\infty$.

What we will in fact show is that for any bounded Borel sets
${\mathcal{A}}\subset{\mathbb{R}}^{2}$ with boundary of Lebesgue measure zero
and any integer $r$,

(9)
$\lim_{T\to\infty}{\mathbb{P}}\big{(}\\#(\Theta_{T}\cap{\mathcal{A}})=r\big{)}={\mathbb{P}}\big{(}\\#(\Theta\cap{\mathcal{A}})=r\big{)}.$

Since $\Theta$ is a simple point process, the convergence in (9) implies the
convergence in distribution asserted in Theorem 1 [6, Theorem 16.16]. An
important point in this argument is the observation that, by Theorem 3 below,
we have $\operatorname{vol}\partial{\mathcal{A}}=0$ if and only if
$\\#(\Theta\cap\partial{\mathcal{A}})=0$ almost surely.

We will see below (Theorem 6) that $\Theta$ is not invariant under the
standard $\operatorname{SL}(2,{\mathbb{R}})$ action on ${\mathbb{R}}^{2}$,
although its two-point function is the same as that of a Poisson process
(Theorem 5). Similar processes arise in the Boltzmann-Grad limit of the
Lorentz gas in polycrystals [13].

Note that Elkies and McMullen [4] do not see the full process $\Theta$ since
their setting corresponds to triangular test sets of the form $\\{(x,y)\mid
0<x<1,\;|y|\leq\sigma x\\}$, which are contained in the right half plane
${\mathbb{H}}_{+}$. By definition (7), $\Theta$ is indistinguishable from the
random affine lattice ${\mathbb{Z}}^{2}g$ when restricted to the right half
plane.

The plan for the proof of Theorem 1 is to show that, in any bounded set
${\mathcal{A}}\subset{\mathbb{R}}^{2}$, the set $\Theta_{T}$ is very close to
an affine lattice, and then apply a slight extension of the Elkies-McMullen
equidistribution theorem, which we will state now.

##  3\. Equidistribution

Let

(10) $N(\xi)=\bigg{(}\begin{pmatrix}1&2\xi\\\
0&1\end{pmatrix},(-\xi,-\xi^{2})\bigg{)}.$

Note that the embedding ${\mathbb{R}}\to G$, $\xi\mapsto N(\xi)$, defines a
group homomorphism.

######  Theorem 2.

Let $\lambda$ be an absolutely continuous probability measure on
${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$. Then, for any bounded continuous
function $f:X\times X\to{\mathbb{R}}$,

(11) $\lim_{T\to\infty}\int_{\mathbb{T}}f\big{(}\Gamma
N(\xi)D(T),\Gamma(1,(\tfrac{1}{2},-\tfrac{1}{4}))N(\xi)D(T)\big{)}d\lambda(\xi)=\int_{Y}f\big{(}\Gamma
g,\Gamma(1,(\tfrac{1}{2},-\tfrac{1}{4}))g\big{)}d\mu(g).$

###### Proof.

The space $Y$ is a finite cover of $X$. The equidistribution theorem of [4]
extends to $Y$ (and in fact to any quotient by a finite-index subgroup of
$\Gamma$): For any bounded continuous function $h:Y\to{\mathbb{R}}$, we have

(12) $\lim_{T\to\infty}\int_{{\mathbb{T}}}h\big{(}\Lambda
N(\xi)D(T)\big{)}d\lambda(\xi)=\int_{Y}h(g)d\mu(g).$

To complete the proof, choose $h(g)=f\big{(}\Gamma
g,\Gamma(1,(\tfrac{1}{2},-\tfrac{1}{4}))g\big{)}$ and note that due to (8)
this $h$ is indeed well-defined and bounded continuous as a function on $Y$. ∎

Effective versions of the equidistribution theorem of [4], which is a
consequence of Ratner’s measure classification theorem [15, 18], have recently
been established in [1].

##  4\. Properties of the limit process

The intensity of a random point set $\Theta$ is defined by

(13)
$I_{f}(\Theta)={\mathbb{E}}\sum_{{\text{\boldmath$y$}}\in\Theta}f({\text{\boldmath$y$}}),$

where $f\in\operatorname{C{}}_{0}({\mathbb{R}}^{2})$, i.e. continuous and with
compact support. The following claim states that the intensity measure of our
process is the Lebesgue measure.

######  Theorem 3.

For $f\in\operatorname{L{}}^{1}({\mathbb{R}}^{2})$,

(14)
$I_{f}(\Theta)=\int_{{\mathbb{R}}^{2}}f({\text{\boldmath$x$}})d{\text{\boldmath$x$}}.$

###### Proof.

The point processes defined by the point sets $\Xi={\mathbb{Z}}^{2}g$,
$\widetilde{\Xi}=-([{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})]g)$, with
$g\in Y$ randomly distributed according to $\mu$, are translation invariant
and have asymptotic density equal to one. ($\Xi$ is in fact the same process
as discussed in the introduction, despite the fact that we are now working
with $\Lambda$ rather than $\operatorname{ASL}(2,{\mathbb{Z}})$.) Therefore,
by Campbell’s formula,

(15)
$I_{f}(\Xi)=\int_{{\mathbb{R}}^{2}}f({\text{\boldmath$x$}})d{\text{\boldmath$x$}},\qquad
I_{f}(\widetilde{\Xi})=\int_{{\mathbb{R}}^{2}}f({\text{\boldmath$x$}})d{\text{\boldmath$x$}}.$

Thus, for
$f_{\pm}({\text{\boldmath$x$}})=f({\text{\boldmath$x$}})\chi_{{\mathbb{H}}_{\pm}}({\text{\boldmath$x$}})$
the restriction of $f$ to the respective half plane,

(16)
$I_{f}(\Theta)=I_{f_{+}}(\Xi)+I_{f_{-}}(\widetilde{\Xi})=\int_{{\mathbb{R}}^{2}}f_{+}({\text{\boldmath$x$}})d{\text{\boldmath$x$}}+\int_{{\mathbb{R}}^{2}}f_{-}({\text{\boldmath$x$}})d{\text{\boldmath$x$}}=\int_{{\mathbb{R}}^{2}}f({\text{\boldmath$x$}})d{\text{\boldmath$x$}}.$

∎

Theorem 3 implies the useful fact that for any Borel set
${\mathcal{A}}\subset{\mathbb{R}}^{2}$ we have
$\operatorname{vol}\partial{\mathcal{A}}=0$ if and only if
$\\#(\Theta\cap\partial{\mathcal{A}})=0$ almost surely. To see this choose $f$
as the indicator function of $\partial{\mathcal{A}}$, which yields
${\mathbb{E}}\,\\#(\Theta\cap\partial{\mathcal{A}})=\operatorname{vol}\partial{\mathcal{A}}$.

Let us now consider second-order correlations. For
$f\in\operatorname{C{}}_{0}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$, define
the two-point function $R_{f}^{\pm}:Y\to{\mathbb{R}}$ by

(17)
$R_{f}^{+}(g)=\sum_{{\text{\boldmath$m$}}_{1}\neq{\text{\boldmath$m$}}_{2}\in{\mathbb{Z}}^{2}}f({\text{\boldmath$m$}}_{1}g,{\text{\boldmath$m$}}_{2}g),\qquad
R_{f}^{-}(g)=\sum_{{\text{\boldmath$m$}}_{1},{\text{\boldmath$m$}}_{2}\in{\mathbb{Z}}^{2}}f({\text{\boldmath$m$}}_{1}g,-[{\text{\boldmath$m$}}_{2}(1,(\tfrac{1}{2},-\tfrac{1}{4}))g]).$

This defines the linear functionals $R_{f}^{\pm}$ from
$\operatorname{L{}}^{1}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$ to
$\operatorname{L{}}^{1}(Y,d\mu)$.

######  Theorem 4.

For $f\in\operatorname{L{}}^{1}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$,

(18)
$\int_{Y}R_{f}^{\pm}(g)d\mu(g)=\int_{{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}}f({\text{\boldmath$x$}},{\text{\boldmath$y$}})d{\text{\boldmath$x$}}d{\text{\boldmath$y$}}.$

###### Proof.

The relation for $R_{f}^{+}$ is proved in [3, Proposition A.3]. The other is
analogous: In view of the density of $\operatorname{C{}}_{0}$ in
$\operatorname{L{}}^{1}$ and the Lebesgue monotone convergence theorem, we may
assume without loss of generality that
$f\in\operatorname{C{}}_{0}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$. Let
$\mu_{0}$ denote the $\operatorname{SL}(2,{\mathbb{R}})$-invariant probability
measure on $Y_{0}=\Gamma_{2,0}(4)\backslash\operatorname{SL}(2,{\mathbb{R}})$.
Then

(19)
$\begin{split}\int_{Y}R_{f}^{-}(g)d\mu(g)&=\int_{Y_{0}}\int_{{\mathbb{T}}^{2}}\sum_{{\text{\boldmath$m$}}_{1},{\text{\boldmath$m$}}_{2}}f(({\text{\boldmath$m$}}_{1}+{\text{\boldmath$y$}})M,-({\text{\boldmath$m$}}_{2}+{\text{\boldmath$y$}}+(\tfrac{1}{2},-\tfrac{1}{4}))M)d{\text{\boldmath$y$}}d\mu_{0}(M)\\\
&=\int_{Y_{0}}\int_{{\mathbb{R}}^{2}}\sum_{{\text{\boldmath$m$}}}f({\text{\boldmath$y$}},-({\text{\boldmath$m$}}+(\tfrac{1}{2},-\tfrac{1}{4}))M-{\text{\boldmath$y$}})d{\text{\boldmath$y$}}d\mu_{0}(M).\end{split}$

The Siegel integral formula for
$\tilde{f}\in\operatorname{C{}}_{0}({\mathbb{R}}^{2})$ reads

(20)
$\int_{Y_{0}}\sum_{{\text{\boldmath$m$}}\in{\mathbb{Z}}^{2}}\tilde{f}(({\text{\boldmath$m$}}+(\tfrac{1}{2},-\tfrac{1}{4}))M)d\mu_{0}(M)=\int_{{\mathbb{R}}^{2}}\tilde{f}({\text{\boldmath$x$}})d{\text{\boldmath$x$}}.$

This follows either by direct computation or the general Siegel-Veech formula
[23]; it is in fact also a special case of the generalized Siegel-Veech
formula for Euclidean model sets [12, Theorem 5.1]. Applying this with

(21)
$\tilde{f}({\text{\boldmath$x$}})=\int_{{\mathbb{R}}^{2}}f({\text{\boldmath$y$}},-{\text{\boldmath$x$}}-{\text{\boldmath$y$}})d{\text{\boldmath$y$}}$

proves the theorem. ∎

The two-point function of the random point process $\Theta$ is defined by

(22)
$K_{f}(\Theta)={\mathbb{E}}\sum_{{\text{\boldmath$y$}}_{1}\neq{\text{\boldmath$y$}}_{2}\in\Theta}f({\text{\boldmath$y$}}_{1},{\text{\boldmath$y$}}_{2}),$

where $f\in\operatorname{C{}}_{0}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$.
The following corollary of Theorem 4 shows that the the two-point function of
$\Theta$ is the same as that of a Poisson process. This extends the
observation of [2] for $\sqrt{n}\bmod 1$ to the full two-dimensional process.

######  Theorem 5.

For $f\in\operatorname{L{}}^{1}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$,

(23)
$K_{f}(\Theta)=\int_{{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}}f({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})d{\text{\boldmath$x$}}_{1}d{\text{\boldmath$x$}}_{2}.$

###### Proof.

Let

(24) $\displaystyle
f_{++}({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})$
$\displaystyle=f({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})\chi_{{\mathbb{H}}_{+}}({\text{\boldmath$x$}}_{1})\chi_{{\mathbb{H}}_{+}}({\text{\boldmath$x$}}_{2}),$
$\displaystyle f_{+-}({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})$
$\displaystyle=f({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})\chi_{{\mathbb{H}}_{+}}({\text{\boldmath$x$}}_{1})\chi_{{\mathbb{H}}_{-}}({\text{\boldmath$x$}}_{2}),$
(25) $\displaystyle
f_{-+}({\text{\boldmath$x$}}_{2},{\text{\boldmath$x$}}_{1})$
$\displaystyle=f({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})\chi_{{\mathbb{H}}_{-}}({\text{\boldmath$x$}}_{1})\chi_{{\mathbb{H}}_{+}}({\text{\boldmath$x$}}_{2}),$
$\displaystyle f_{--}({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})$
$\displaystyle=f({\text{\boldmath$x$}}_{1},{\text{\boldmath$x$}}_{2})\chi_{{\mathbb{H}}_{-}}({\text{\boldmath$x$}}_{1})\chi_{{\mathbb{H}}_{-}}({\text{\boldmath$x$}}_{2}),$

be the restrictions to the various half planes. We have

(26)
$K_{f}(\Theta)=\int_{Y}\big{[}R_{f_{++}}^{+}(g)+R_{f_{--}}^{+}((1,(\tfrac{1}{2},-\tfrac{1}{4}))g)+R_{f_{+-}}^{-}(g)+R_{f_{-+}}^{-}(g)\big{]}d\mu(g).$

The corollary now follows from Theorem 4 by integrating term-wise. ∎

Given a subgroup $H\subset G$, we say $\Theta$ is $H$-invariant if $\Theta h$
has the same distribution as $\Theta$ (viewed as random point sets in
${\mathbb{R}}^{2}$) for all $h\in H$.

Consider the subgroup

(27) $P=\bigg{\\{}\bigg{(}\begin{pmatrix}a&b\\\
0&1/a\end{pmatrix},(0,y)\bigg{)}\,\bigg{|}\,b,y\in{\mathbb{R}},\;a\in{\mathbb{R}}\setminus\\{0\\}\bigg{\\}}\subset
G.$

######  Theorem 6.

$\Theta$ is $P$-invariant but not
$\operatorname{SL}(2,{\mathbb{R}})$-invariant.

###### Proof.

The invariance under

(28) $\bigg{(}\begin{pmatrix}a&b\\\ 0&1/a\end{pmatrix},(0,y)\bigg{)}$

is evident for all $a>0$, $b\in{\mathbb{R}}$, since these transformations
preserve the half planes ${\mathbb{H}}_{+}$ and ${\mathbb{H}}_{-}$. What
remains for the proof of $P$-invariance is the invariance under the element
$(-1,(0,0))$, which acts on ${\mathbb{R}}^{2}$ by reflection at the origin. By
the $G$-invariance of $\mu$, $\Gamma g$ has the same distribution as
$\Gamma(-1,(\tfrac{1}{2},-\tfrac{1}{4}))g$. This implies that
$\Theta=\big{(}{\mathbb{Z}}^{2}g\cap{\mathbb{H}}_{+}\big{)}\cup\big{(}-\big{(}\big{[}{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})\big{]}g\big{)}\cap{\mathbb{H}}_{-}\big{)}$
has the same distribution as the random point set

(29)
$\begin{split}\Theta^{\prime}&=\big{(}{\mathbb{Z}}^{2}(-1,(\tfrac{1}{2},-\tfrac{1}{4}))g\cap{\mathbb{H}}_{+}\big{)}\cup\big{(}-\big{(}\big{[}{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})\big{]}(-1,(\tfrac{1}{2},-\tfrac{1}{4}))g\big{)}\cap{\mathbb{H}}_{-}\big{)}\\\
&=\big{(}\big{[}{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})\big{]}g\cap{\mathbb{H}}_{+}\big{)}\cup\big{(}-\big{(}{\mathbb{Z}}^{2}g\big{)}\cap{\mathbb{H}}_{-}\big{)}.\end{split}$

This shows $\Theta=-\Theta^{\prime}$, which establishes the
$(-1,(0,0))$-invariance.

As to the $\operatorname{SL}(2,{\mathbb{R}})$-invariance, we note that every
realisation of $\Theta$, restricted to the right half planes
${\mathbb{H}}_{+}$, looks like a affine lattice ${\mathbb{Z}}^{2}$ restricted
to the right half plane. It is evident that the rotated process

$\widetilde{\Theta}=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\Theta$

does not have this property, since every realisation produces two different
affine lattices (which are copies of the same underlying lattice) in the upper
and lower half plane, respectively. This shows that $\widetilde{\Theta}$ and
$\Theta$ do not have the same distribution. Hence $\Theta$ is not
$\operatorname{SL}(2,{\mathbb{R}})$-invariant. ∎

##  5\. Proof of Theorem 1

To establish the convergence $\Theta_{T}\Rightarrow\Theta$, it is sufficient
to prove that convergence holds in finite-dimensional distribution [6, Theorem
16.16] (recall the remark following Theorem 3). Since $\Theta$ is a simple
point process, it is in fact sufficient to consider the one-dimensional
distributions. That is, we need to prove that for any bounded Borel set
${\mathcal{A}}\subset{\mathbb{R}}^{2}$ with boundary of Lebesgue measure zero,
the random variable $\\#(\Theta_{T}\cap{\mathcal{A}})$ converges in
distribution to $\\#(\Theta\cap{\mathcal{A}})$. It is in fact sufficient to
prove the convergence for test sets ${\mathcal{A}}$ that are rectangles of the
form $[a,b]$. More general Borel sets ${\mathcal{A}}$ (bounded with boundary
of Lebesgue measure zero) can then be approximated by unions of such
rectangles. This will require proof of convergence for the joint distribution
for finitely many rectangles. We will limit the presentation to one rectangle;
the case of multiple rectangles is analogous.

The right half plane. The following estimates are almost identical to those in
the paper [4], which considers the case of triangular test sets of the form
$\\{(x,y)\mid 0<x<1,\;|y|\leq\sigma x\\}$, rather than general rectangles.
Consider a rectangle $[a,b]\times[c,d]$ and assume for now $a\geq 0$. Note
first of all that in (4) the sine has to be of order $(Tn)^{-1/2}$, and thus
its argument must be close, by the same order, to $0$ or $\pi$ mod $2\pi$. If
it is close to $\pi$, the cosine is negative which is ruled out by the
assumption $a\geq 0$. Set $\xi=-\theta/2\pi$, and define $m\in{\mathbb{Z}}$ so
that $-\frac{1}{2}\leq\sqrt{n}+\xi+m<\frac{1}{2}$. Using $4|x|\leq|\sin(2\pi
x)|$ for $|x|\leq\frac{1}{4}$, we have for $Tn$ sufficiently large,

(30)
$4|\sqrt{n}+\xi+m|\leq|\sin(2\pi(\sqrt{n}+\xi))|\leq\max\\{|c|,|d|\\}\sqrt{\frac{\pi}{Tn}}.$

The objective is now to linearise the inequalities

(31) $a\leq\sqrt{\frac{n}{\pi T}}\cos(2\pi(\sqrt{n}+\xi))\leq b,\qquad
c\leq\sqrt{\frac{Tn}{\pi}}\sin(2\pi(\sqrt{n}+\xi))\leq d.$

Taylor’s Theorem tells us that $|\cos(x)-1|\leq\tfrac{1}{2}x^{2}$ and
$|\sin(x)-x|\leq\tfrac{1}{6}|x|^{3}$, and so

(32) $a\leq\sqrt{\frac{n}{\pi
T}}+O\bigg{(}\frac{1}{T^{3/2}n^{1/2}}\bigg{)}\leq b,$ (33)
$\frac{c}{2\sqrt{\pi
Tn}}\leq\sqrt{n}+\xi+m+O\bigg{(}\frac{1}{(Tn)^{3/2}}\bigg{)}\leq\frac{d}{2\sqrt{\pi
Tn}}.$

The last inequality transforms to

(34) $\frac{c}{2\sqrt{\pi
Tn}}-(m+\xi)\leq\sqrt{n}+O\bigg{(}\frac{1}{(Tn)^{3/2}}\bigg{)}\leq\frac{d}{2\sqrt{\pi
Tn}}-(m+\xi),$

which is equivalent to

(35) $\bigg{[}\frac{c}{2\sqrt{\pi Tn}}-(m+\xi)\bigg{]}^{2}\leq
n+O\bigg{(}\frac{1}{T^{3/2}n}\bigg{)}\leq\bigg{[}\frac{d}{2\sqrt{\pi
Tn}}-(m+\xi)\bigg{]}^{2}.$

Now (35) is equivalent to

(36) $-\frac{c(m+\xi)}{\sqrt{\pi Tn}}+(m+\xi)^{2}\leq
n+O\bigg{(}\frac{1}{Tn}\bigg{)}\leq-\frac{d(m+\xi)}{\sqrt{\pi
Tn}}+(m+\xi)^{2}.$

Using $\sqrt{n}=-(m+\xi)+O(1/\sqrt{Tn})$ we get that

(37) $\frac{c}{\sqrt{\pi T}}+(m+\xi)^{2}\leq
n+O\bigg{(}\frac{1}{Tn}\bigg{)}\leq\frac{d}{\sqrt{\pi T}}+(m+\xi)^{2},$

and also from (32)

(38) $a\leq-\frac{m+\xi}{\sqrt{\pi T}}+O\bigg{(}\frac{1}{Tn^{1/2}}\bigg{)}\leq
b.$

Let us assume that $\xi\notin[-\delta,\delta]+{\mathbb{Z}}$, where $\delta>0$
may be chosen arbitrarily small. (This assumption is without loss of
generality, because the event $\xi\in[-\delta,\delta]+{\mathbb{Z}}$ has
probability at most $\lambda([-\delta,\delta])$, which tends to zero as
$\delta\to 0$.) We can then drop the condition $n\geq 1$, as the positivity of
$n$ is implied in (37) for $T$ sufficiently large. Next, replacing $n$ by
$n+m^{2}$ yields

(39) $a\leq-\frac{m+\xi}{\sqrt{\pi T}}+O\bigg{(}\frac{1}{T}\bigg{)}\leq
b,\qquad\frac{c}{\sqrt{\pi T}}\leq
n-2m\xi-\xi^{2}+O\bigg{(}\frac{1}{T}\bigg{)}\leq\frac{d}{\sqrt{\pi T}}.$

So, given any $\epsilon>0$ there exists $T>0$ such that the number of points
$(m,n)\in{\mathbb{Z}}^{2}$ satisfying (39) is bounded above (resp. below) by
the number of points with

(40) $\bigg{(}-\frac{m+\xi}{\sqrt{\pi T}},\sqrt{\pi
T}(n-2m\xi-\xi^{2})\bigg{)}\in\text{${\mathcal{A}}_{\epsilon}^{+}$ (resp.\
${\mathcal{A}}_{\epsilon}^{-}$)}$

with

(41)
${\mathcal{A}}_{\epsilon}^{+}=[a-\epsilon,b+\epsilon]\times[c-\epsilon,d+\epsilon],\qquad{\mathcal{A}}_{\epsilon}^{-}=[a+\epsilon,b-\epsilon]\times[c+\epsilon,d-\epsilon].$

Note that (40) can be expressed as

(42) $(-m,n)N(\xi)D(\pi T)\in\text{${\mathcal{A}}_{\epsilon}^{+}$ (resp.\
${\mathcal{A}}_{\epsilon}^{-}$)}.$

We have thus established that in rectangles with $a\geq 0$ the process
$\Theta_{T}$ looks asymptotically like the random affine lattice

(43) $\Xi_{T}={\mathbb{Z}}^{2}N(\xi)D(\pi T),$

see Figure 3. We consider $\Xi_{T}$ as a process in ${\mathbb{R}}^{2}$, so the
fact that ${\mathcal{A}}_{\epsilon}^{+}$ might intersect with the left half
plane (e.g. in the case $a=0$) is not a cause for concern. Elkies and McMullen
have shown that for $\xi$ random with respect to any absolutely continuous
probability measure on ${\mathbb{R}}/{\mathbb{Z}}$, the process $\Xi_{T}$
converges in distribution to $\Xi$, as $T\to\infty$. In view of (15), the
probability of having one or more points in
${\mathcal{A}}_{\epsilon}^{+}\setminus{\mathcal{A}}_{\epsilon}^{-}$ is of
order $\epsilon$. The limits $T\to\infty$ and $\epsilon\to 0$ therefore
commute. This would complete the proof if we were only interested in test sets
in the right half plane.

The left half plane. Let us therefore turn to the case $b\leq 0$. In this case
the argument of the sine has to be close to $\pi$ as the cosine is now
required to be negative. Then (33) is replaced by

(44) $-\frac{d}{2\sqrt{\pi
Tn}}\leq\sqrt{n}+\zeta+m+O\bigg{(}\frac{1}{(Tn)^{3/2}}\bigg{)}\leq-\frac{c}{2\sqrt{\pi
Tn}}.$

with $\zeta=-(\theta+\pi)/2\pi=\xi-1/2$. Repeating the steps in the previous
calculation leads to

(45) $a\leq\frac{m+\zeta}{\sqrt{\pi T}}+O\bigg{(}\frac{1}{T}\bigg{)}\leq
b,\qquad\frac{c}{\sqrt{\pi
T}}\leq-n+2m\zeta+\zeta^{2}+O\bigg{(}\frac{1}{T}\bigg{)}\leq\frac{d}{\sqrt{\pi
T}}.$ (46) $-\frac{d}{\sqrt{\pi T}}\leq
n-2m\zeta-\zeta^{2}+O\bigg{(}\frac{1}{T}\bigg{)}\leq-\frac{c}{\sqrt{\pi
T}},\qquad a\sqrt{\pi T}\leq m+\zeta+O\bigg{(}\frac{1}{T}\bigg{)}\leq
b\sqrt{\pi T}.$

So upper (resp. lower) bounds on the number of points are given by

(47) $\bigg{(}\frac{m+\zeta}{\sqrt{\pi T}},\sqrt{\pi
T}(-n+2m\zeta+\zeta^{2})\bigg{)}\in\text{${\mathcal{A}}_{\epsilon}^{+}$
(resp.\ ${\mathcal{A}}_{\epsilon}^{-}$)},$

and hence

(48) $(m,-n)\bigg{(}\begin{pmatrix}1&2\zeta\\\
0&1\end{pmatrix},(\zeta,\zeta^{2})\bigg{)}D(\pi
T)\in\text{${\mathcal{A}}_{\epsilon}^{+}$ (resp.\
${\mathcal{A}}_{\epsilon}^{-}$)}.$

We have thus shown that in rectangles with $b\leq 0$ the process $\Theta_{T}$
looks like the affine lattice

(49) ${\mathbb{Z}}^{2}\bigg{(}\begin{pmatrix}1&2\zeta\\\
0&1\end{pmatrix},(\zeta,\zeta^{2})\bigg{)}D(\pi
T)=-\big{[}{\mathbb{Z}}^{2}N(\zeta)D(\pi T)\big{]}.$

Finally we check that

(50) ${\mathbb{Z}}^{2}N(\zeta)={\mathbb{Z}}^{2}\bigg{(}\begin{pmatrix}1&-1\\\
0&1\end{pmatrix},(\tfrac{1}{2},-\tfrac{1}{4})\bigg{)}N(\xi)\\\
={\mathbb{Z}}^{2}\big{(}1,(\tfrac{1}{2},-\tfrac{1}{4})\big{)}N(\xi).$

We have now established that in rectangles with $b\leq 0$ the process
$\Theta_{T}$ looks asymptotically like the random affine lattice

(51)
$\widetilde{\Xi}_{T}=-[{\mathbb{Z}}^{2}\big{(}1,(\tfrac{1}{2},-\tfrac{1}{4})\big{)}N(\xi)D(\pi
T)],$

see Figure 4. As in the case of the right half plane, the process
$\widetilde{\Xi}_{T}$ converges in distribution to $\Xi$, as $T\to\infty$,
provided we restrict test sets to rectangles in the left half plane. The final
challenge is now to combine these results to establish joint convergence in
both half planes.

Joint convergence. Let us now consider the remaining case $a\leq 0\leq b$. We
decompose the rectangle as

(52) $[a,b]\times[c,d]=([a,0]\times[c,d])\cup([0,b]\times[c,d]).$

The argument for the left and right halfnplanes show that an upper bound for
the number of points is obtained by considering, for any $\epsilon>0$, the
joint distribution of

(53)
$\widetilde{\Xi}_{T}\cap([a-\epsilon,\epsilon]\times[c-\epsilon,d+\epsilon]),\qquad\Xi_{T}\cap([-\epsilon,b+\epsilon]\times[c-\epsilon,d+\epsilon]),$

where
$\widetilde{\Xi}_{T}={\mathbb{Z}}^{2}\big{(}1,(\tfrac{1}{2},-\tfrac{1}{4})\big{)}N(\xi)D(\pi
T)$ and $\Xi_{T}={\mathbb{Z}}^{2}N(\xi)D(\pi T)$ are not independent, since
the are functions of the same random varaible $\xi$. Theorem 2 implies that
the joint limit distribution of (53) is given by

(54)
$\widetilde{\Xi}\cap([a-\epsilon,\epsilon]\times[c-\epsilon,d+\epsilon]),\qquad\Xi\cap([-\epsilon,b+\epsilon]\times[c-\epsilon,d+\epsilon]),$

where $\widetilde{\Xi}=[{\mathbb{Z}}^{2}+(\tfrac{1}{2},-\tfrac{1}{4})]g$,
$\Xi={\mathbb{Z}}^{2}g$, and $g\in Y$ randomly distributed according to $\mu$.
Similarly, a lower bound on the number of points is given by

(55)
$\widetilde{\Xi}\cap([a+\epsilon,-\epsilon]\times[c+\epsilon,d-\epsilon]),\qquad\Xi\cap([\epsilon,b-\epsilon]\times[c+\epsilon,d-\epsilon]).$

The remark following Theorem 3 provides the regularity that allows us to
exchange the limits $T\to\infty$ and $\epsilon\to 0$. This shows that indeed
$\Theta_{T}\cap([a,b]\times[c,d])$ converges in distribution to

(56)
$[\widetilde{\Xi}\cap([a,0]\times[c,d])]\cup[\Xi\cap([0,b]\times[c,d])]=\Theta\cap([a,b]\times[c,d]),$

as required. ∎

## References

  * [1] T. Browning and I. Vinogradov, Effective Ratner theorem for $\operatorname{ASL}(2,{\mathbb{R}})$ and gaps in $\sqrt{n}$ modulo 1, J. London Math. Soc. 94 (2016), 61–84.
  * [2] D. El-Baz, J. Marklof and I. Vinogradov, The two-point correlation function of the fractional parts of $\sqrt{n}$ is Poisson, Proc. AMS 143 (2015), 2815–2828
  * [3] D. El-Baz, J. Marklof and I. Vinogradov, The distribution of directions in an affine lattice: two-point correlations and mixed moments. IMRN 2015, no. 5, 1371–1400.
  * [4] N.D. Elkies and C.T. McMullen, Gaps in $\sqrt{n}\bmod 1$ and ergodic theory. Duke Math. J. 123 (2004), 95–139.
  * [5] K. Fraczek, R. Shi and C. Ulcigrai, Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions, J. Mod. Dyn. 12 (2018), 55–122.
  * [6] O. Kallenberg, Foundations of modern probability, 2nd Edition, Springer-Verlag, New York, 2002.
  * [7] W. Kim and J. Marklof, Poissonian pair correlation for directions in multi-dimensional affine lattices, and escape of mass estimates for embedded horospheres, Ergodic Theory Dynam. Systems, to appear; arXiv:2302.13308
  * [8] C. Lutsko, A. Sourmelidis and N. Technau, Pair correlation of the fractional parts of $\alpha n^{\theta}$, J. Eur. Math. Soc., to appear; arXiv:2106.09800
  * [9] J. Marklof, Distribution modulo one and Ratner’s theorem, Equidistribution in Number Theory, An Introduction, eds. A. Granville and Z. Rudnick, Springer 2007, pp. 217–244.
  * [10] J. Marklof, Delone sets generated by square roots, Am. Math. Monthly 127 (2020) 836–840.
  * [11] J. Marklof and A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Annals of Math. 172 (2010), 1949–2033.
  * [12] J. Marklof and A. Strömbergsson, Free path lengths in quasicrystals, Comm. Math.Phys. 330 (2014), 723–755.
  * [13] J. Marklof and A. Strömbergsson, Generalized linear Boltzmann equations for particle transport in polycrystals. Appl. Math. Res. Express. AMRX 2015, no. 2, 274–295.
  * [14] J. Marklof and I. Vinogradov, Spherical averages in the space of marked lattices, Geometriae Dedicata 186 (2017) 75–102.
  * [15] D.W. Morris, Ratner’s theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005.
  * [16] S. Pattison, Rational points on nonlinear horocycles and pigeonhole statistics for the fractional parts of $\sqrt{n}$, Ergodic Theory Dynam. Systems 43 (2023) 3108–3130.
  * [17] M. Radziwiłł and N. Technau, Gap distribution of $\sqrt{n}$ mod 1 and the circle method, arXiv:2403.16493
  * [18] M. Ratner, On Raghunathan’s measure conjecture, Annals of Math. 134 (1991) 545–607.
  * [19] Ya. G. Sinai, Statistics of gaps in the sequence $\\{\sqrt{n}\\}$. In Dynamical Systems and Group Actions, volume 567 of Contemp. Math., pages 185–189. Amer. Math. Soc., Providence, RI, 2012.
  * [20] D. Stoyan, W.S. Kendall and J. Mecke, Stochastic Geometry and its Applications, 2nd edition, John Wiley & Sons, Chichester, 1995.
  * [21] A. Strömbergsson, An effective Ratner equidistribution result for $\operatorname{SL}(2,{\mathbb{R}})\ltimes{\mathbb{R}}^{2}$. Duke Math. J. 164 (2015), no. 5, 843–902.
  * [22] N. Technau and N. Yesha, On the correlations of $n^{\alpha}\bmod 1$, J. Eur. Math. Soc. 25 (2023), no.10, 4123–4154.
  * [23] W. A. Veech, Siegel measures, Ann. of Math. 148 (1998), 895–944.