[article] [abstract] 

J.S. Athreya, G. Margulis
Logarithm Laws for Unipotent Flows, II.
preprint, 2014.
Abstract:
We prove analogs of the logarithm laws of Sullivan and KleinbockMargulis
in the context of unipotent flows. In particular, we prove results for
horospherical actions on homogeneous spaces $G/\Gamma$. We describe some
relations with multidimensional diophantine approximation.


[article] [abstract] 

J.S. Athreya
Random Affine Lattices.
Contemporary Mathematics,
volume 639, 169174, 2015.
Abstract:
We give a bound on the probability that a randomly chosen affine
unimodular lattice has large holes, and a similar bound on the probability
of large holes in the spectrum of a random flat torus. We discuss various
motivations and generalizations, and state several open questions.


[article] [abstract] 

J.S. Athreya, C.Cobeli, A. Zaharescu
Radial Density in Apollonian Packings.
International
Mathematics Research Notices, 2014.
Abstract:
Given an Apollonian Circle Packing $\mathcal{P}$, and a circle $C_0
= \partial B(z_0, r_0)$ in $\mathcal{P}$, color the set of disks in
$\mathcal{P}$
tangent to $C_0$ red. What proportion of the concentric circle
$C_{\epsilon} =
\partial B(z_0, r_0 + \epsilon)$ is red, and what is the behavior of this
quantity as $\epsilon \rightarrow 0$?
Using equidistribution of closed horocycles on the modular surface
$\mathbb{H}^2/SL(2, \mathbb{Z})$, we show that the answer is
$\frac{3}{\pi} =
0.9549\dots$ We also describe an observation due to Alex Kontorovich
connecting the rate of this convergence in the FareyFord packing to the
Riemann Hypothesis. For the analogous problem for Soddy Sphere packings,
we find that
the limiting radial density is $\frac{\sqrt{3}}{2V_T}=0.853\dots$, where
$V_T$
denotes the volume of an ideal hyperbolic tetrahedron.


[article] [abstract] 

J.S. Athreya, S. Chaubey, A. Malik, A. Zaharescu
Geometric statistics of Ford circles.
New York Journal of Mathematics 21 (2015) 637–656.
Abstract:
The Farey sequence is a natural exhaustion of the set of
rational numbers between 0 and 1 by finite lists. Ford Circles are a
natural family of mutually tangent circles associated to Farey fractions:
they are an important object of study in the geometry of numbers and
hyperbolic geometry. We define two sequences of polygons associated to
these objects, the Euclidean and hyperbolic Farey–Ford polygons. We
study the asymptotic behavior of these polygons by exploring various
geometric properties such as (but not limited to) areas, length and slopes
of sides, and angles between sides


[article] [abstract] 

J. S. Athreya, A. Prasad
Growth in RightAngled Artin Groups and Monoids.
preprint, 2014.
Abstract:
We derive functional relationships between spherical generating functions
of graph monoids, rightangled Artin groups and rightangled Coxeter
groups. We use these relationships to express the spherical generating
function of a rightangled Artin group in terms of the clique polynomial
of its defining graph. We also describe algorithms for computing the
geodesic generating functions of these structures.


[article] [abstract] 

J. S. Athreya, A. Ghosh, J. Tseng
Spherical averages of Siegel transforms for higher rank diagonal
actions and applications
preprint, 2014.
Abstract:
We investigate the geometry of approximates in multiplicative Diophantine
approximation. Our main tool is a new multiparameter averaging result for
Siegel transforms on the space of unimodular lattices in $\mathbb R^n$
which is of independent interest.


[article]
[abstract] 

J. S. Athreya, J. Chaika
The Hausdorff Dimension of NonUniquely Ergodic directions in H(2) is
almost everywhere 1/2. Geometry & Topology 19 (2015) 3537–3563.
Abstract:
We show that for almost every (with respect to MasurVeech measure)
$\omega
\in \mathcal{H}(2)$, the set of angles $\theta \in [0, 2\pi)$ so that
$e^{i\theta}\omega$ has nonuniquely ergodic vertical foliation has
Hausdorff
dimension (and codimension) $1/2$.


[article] [abstract] 

J. S. Athreya, A. Parrish, J. Tseng
Ergodic Theory and Diophantine approximation for linear forms and
translation surfaces.
Nonlinearity, Volume 29, Number 8
, 2016.
Abstract:
We give a simple proof of a version of a classical theorem in
multidimensional Diophantine approximation due to W. Schmidt. While our
version is weaker, the proof relies only on the Birkhoff ergodic theorem
and the Siegel mean value theorem. Our technique also yields results on
systems of linear forms and gives us an analogous result in the setting of
translation surfaces.


[article] [abstract]


J. S. Athreya, J. Chaika, S. Lelievre
The gap distribution of slopes on the golden L.
Contemporary Mathematics,
volume 631, 4762, 2015.
Abstract:
We give an explicit formula for the limiting gap distribution of slopes of
saddle connections on the golden L, or any translation surface in its
SL(2, $\mathbb{R}$)orbit, in particular the double pentagon. This is the
first explicit computation of the distribution of gaps for a flat surface
that is not a torus cover.


[article] [abstract] 

J. S. Athreya, A. Ghosh, J. Tseng
Spherical averages of Siegel transforms and spiraling of lattice
approximations.
Journal of London Mathematical Society, (2015) 91 (2): 383404.
Abstract:
We adapt a very recent proof of Marklof and Strombergsson to show a
spherical average result for Siegel transforms on $\SL_{d+1}(\mathbb
R)/SL_{d+1}(\mathbb Z)$. Our techniques are elementary. Results like this
date back to the work of EskinMargulisMozes and have wideranging
applications. We explore a particular application of our spherical average
result relating to the spiraling of lattice approximates. We show that, on
average, the directions of approximates spiral in a uniformly distributed
fashion on the d?1 dimensional unit sphere. We also obtain results for the
approximation of higherdimensional subspaces. On the other hand, we
explicitly construct examples in which the directions are not uniformly
distributed.


[article] [abstract]


J. S. Athreya, A. Eskin, A. Zorich
Counting generalized JenkinsStrebel differentials.
Geometriae
Dedicata, Volume 170, 195217, 2014.
Abstract:
We study the combinatorial geometry of "lattice" JenkinsStrebel
differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and
of the corresponding counting functions. Developing the results of M.
Kontsevich we evaluate the leading term of the symmetric polynomial
counting the number of such "lattice" JenkinsStrebel differentials having
all zeroes on a single singular layer. This allows us to express the
number of general "lattice" JenkinsStrebel differentials as an
appropriate weighted sum over decorated trees.
The problem of counting JenkinsStrebel differentials is equivalent to the
problem of counting pillowcase covers, which serve as integer points in
appropriate local coordinates on strata of moduli spaces of meromorphic
quadratic differentials. This allows us to relate our counting problem to
calculations of volumes of these strata . A very explicit expression for
the volume of any stratum of meromorphic quadratic differentials recently
obtained by the authors leads to an interesting combinatorial identity for
our sums over trees.


[article] [abstract]


J. S. Athreya, A. Eskin, A. Zorich, with an appendix by J. Chaika
Rightangled billiards and volumes of moduli spaces of quadratic
differentials on $\mathbb{C}P^1$.
Annales scientifiques ENS, volume 49, number 6, 2016.
Abstract:
We use the relation between the volumes of the strata of meromorphic
quadratic differentials with at most simple poles on the Riemann sphere
and counting functions of the number of (bands of) closed geodesics in
associated flat metrics with singularities to prove a very explicit
formula for the volume of each such stratum conjectured by M. Kontsevich a
decade ago.
The proof is based on a formula for the Lyapunov exponents of the geodesic
flow on the moduli space, which gives a recursive formula from which the
volumes could be recovered.
An appendix with an ergodic theorem proved by Jon Chaika is added to the
current version. Applying this ergodic theorem to the Teichmueller
geodesic flow we obtain EXACT quadratic asymptotics for the number of
(bands of) closed trajectories and for the number of generalized diagonals
in almost all rightangled billiards. All coefficients in the asymptotics
(expressed in terms of the associated SiegelVeech constants) are
explicitly computed.


[article] [abstract]


J. S. Athreya
Gap distributions and homogeneous dynamics.
Proceedings of ICM Satellite Conference on Geometry, Topology, and
Dynamics in Negative Curvature, forthcoming.
Abstract:
We survey the use of dynamics of $SL(2, \mathbb{R})$actions to understand
gap distributions for various sequences of subsets of [0,1), particularly
those arising from special trajectories of various twodimensional
dynamical systems. We state and prove an abstract theorem that gives a
unified explanation for some of the examples we present.


[article] [abstract] 

J. S. Athreya, M. Boshernitzan
Ergodic Properties of Compositions of Interval Exchange Maps and
Rotations.
Nonlinearity
Volume 26, 417423, 2013.
Abstract:
We study the ergodic properties of compositions of interval exchange
transformations and rotations. We show that for any interval exchange
transformation $T$, there is a full measure set of $\alpha \in [0, 1)$ so
that $T$ composed with $R_{\alpha}$ is uniquely ergodic, where
$R_{\alpha}$ is rotation by $\alpha$.


[article] [abstract]


J. S. Athreya, A. Ghosh, A. Prasad
Buildings, Extensions, and Volume Growth Entropy.
New York Journal of
Mathematics, Volume 19, 111, 2013.
Abstract:
Let F be a nonArchimedean local field and let E be a finite extension of
F. Let G be a split semisimple F group. We discuss how to compare volumes
on the BruhatTits buildings $B_E$ and $B_F$ of $G(E)$ and $G(F)$
respectively.


[article] [abstract] 

J. S. Athreya, Y. Cheung
A Poincaré section for horocycle flow on the space of lattices.
International
Mathematics Research Notices, Issue 10, 26432690, 2014.
Abstract:
We construct a Poincaré section for the horocycle flow on the modular
surface $SL(2, \mathbb R)/SL(2, \mathbb Z)$, and study the associated
first return map, which coincides with a transformation [the
BocaCobeliZaharescu (BCZ) map] defined by Boca et al. We classify
ergodic invariant measures for this map and prove equidistribution of
periodic orbits. As corollaries, we obtain results on the average depth of
cusp excursions and on the distribution of gaps for Farey sequences and
slopes of lattice vectors


[article] [abstract] 

J.S. Athreya, F. Paulin
Logarithm laws for strong unstable foliations in negative curvature and
nonArchimedian Diophantine approximations.
Groups,
Geometry, and Dynamics, Volume 8, Issue 2, 285309, 2014.
Abstract:
Given for instance a finite volume negatively curved Riemannian manifold
M, we give a precise relation between the logarithmic growth rates of the
excursions into cusps neighborhoods of the strong unstable leaves of
negatively recurrent unit vectors of M and their linear divergence rates
under the geodesic flow. As an application to nonArchimedian Diophantine
approximation in positive characteristic, we relate the growth of the
orbits of lattices under oneparameter unipotent subgroups of GL(2, K)
with approximation exponents and continued fraction expansions of elements
of the field K of formal Laurent series over a finite field.


[article] [abstract] 

J.S. Athreya
Cusp Excursions on Parameter Spaces.
Journal
of London Mathematical Society, Volume 87, Issue 3, 741765, 2013.
Abstract:
We prove several results for dynamics of $SL(d, \mathbb{R})$actions on
noncompact parameter spaces by studying associated discrete sets in
Euclidean spaces. This allows us to give elementary proofs of logarithm
laws for horocycle flows on hyperbolic surfaces and moduli spaces of flat
surfaces. We also give applications to quantitative equidistribution and
Diophantine approximation.


[article] [abstract] 

J.S. Athreya, A. Ghosh, A. Prasad
Ultrametric Logarithm Laws, II.
Monatshefte
für Mathematik, Volume 167, 333356, 2012.
Abstract:
We prove positive characteristic versions of the logarithm laws of
Sullivan and KleinbockMargulis and obtain related results in Metric
Diophantine Approximation.


[article] [abstract] 

J.S. Athreya, J. Chaika
The distribution of gaps for saddle connection directions.
Geometric and Functional
Analysis,Volume 22, Issue 6, 14911516, 2012.
Abstract:
Motivated by the study of billiards in polygons, we prove fine results for
the distribution of gaps of directions of saddle connections on
translation surfaces. As an application we prove that for almost every
holomorphic differential $\omega$ on a Riemann surface of genus $g\geq 2$
the smallest gap between saddle connection directions of length at most a
fixed length decays faster than quadratically in the length. We also
characterize the exceptional set: the decay rate is not faster than
quadratic if and only if $\omega$ is a lattice surface.


[article] [abstract] 

J.S. Athreya, A. Bufetov, A. Eskin, M. Mirzakhani
Lattice point asymptotics and volume growth on Teichmuller space.
Duke Mathematical
Journal, Volume 161, Number 6, 10551111, 2012.
Abstract:
We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to
Teichmuller space. Let $x$ be a point in Teichmuller space, and let
$B_R(x)$ be the ball of radius $R$ centered at $x$ (with distances
measured in the Teichmuller metric).
We obtain asymptotic formulas as $R$ tends to infinity for the volume of
$B_R(x)$, and also for for the cardinality of the intersection of $B_R(x)$
with an orbit of the mapping class group.


[article] [abstract] 

J.S. Athreya, G. Margulis
Logarithm laws for unipotent flows, I.
Journal
of Modern Dynamics, Issue 3, 359  378, 2009.
Abstract:
We prove analogues of the logarithm laws of Sullivan and
KleinbockMargulis in the context of unipotent flows. In particular, we
obtain results for oneparameter actions on the space of lattices $SL(n,
\mathbb R)/SL(n, \mathbb Z)$. The key lemma for our results says the
measure of the set of unimodular lattices in $\mathbb R^n$ that does not
intersect a `large' volume subset of $\mathbb R^n$ is `small'. This can be
considered as a random analogue of the classical Minkowski theorem in the
geometry of numbers.


[article] [abstract] 

J.S. Athreya
Logarithm laws and shrinking target properties.
Proceedings
of the Indian Academy of SciencesMathematical Sciences, Volume 119,
Number 4, pages 541559, 2009.
Abstract:
We survey some of the recent developments in the study of logarithm laws
and shrinking target properties for various families of dynamical systems.
We discuss connections to geometry, diophantine approximation, and
probability theory.


[article] [abstract] 

J.S. Athreya, A. Ghosh, A. Prasad
Ultrametric Logarithm Laws, I.
Discrete
and Continuous Dynamical Systems, Volume 2, Issue 2, 337  348,
2009.
Abstract:
We announce ultrametric analogues of the results of KleinbockMargulis for
shrinking target properties of semisimple group actions on symmetric
spaces. The main applications are $S$arithmetic Diophantine approximation
results and logarithm laws for buildings, generalizing the work of
HersonskyPaulin on trees.


[article] [abstract] 

J.S. Athreya, G. Forni
Deviation of ergodic averages for rational polygonal billiards.
Duke Mathematical
Journal, Volume 144, Number 2, 285319, 2008.
Abstract:
We prove a polynomial upper bound on the deviation of ergodic averages for
almost all directional flows on every translation surface, in particular,
for the generic directional flow of billiards in any Euclidean polygon
with rational angles.


[article] [abstract] 

J.S. Athreya
Quantitative Recurrence and Large Deviations for Teichmuller Geodesic
Flow.
Geometriae
Dedicata, Volume 119, Issue 1, pp 121140, 2006.
Abstract:
We prove quantitative recurrence and large deviations results for the
Teichmuller geodesic flow on connected components of strata of the moduli
space $Q_g$ of holomorphic unitarea quadratic differentials on a compact
genus $g$ surface.
