16 citations · 83 across the 21 of their papers we have counts for
25 papers · 1 filter
On the rate of convergence of continued fraction statistics of random rationals
Ofir David, Taehyeong Kim, Ron Mor +1
We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator , approaches the Gauss-Kuzmin…
K-divergent lattices
Guy Lachman, Anurag Rao, Uri Shapira +1
We introduce a novel concept in topological dynamics, referred to as -divergence, which extends the notion of divergent orbits. Motivated by questions in the theory of inhomogen…
Translates of S-arithmetic orbits and applications
Uri Shapira, Cheng Zheng
We prove that certain sequences of periodic orbits of the diagonal group in the space of lattices equidistribute. As an application we obtain new information regarding the sequence…
Translates of rational points along expanding closed horocycles on the modular surface
Claire Burrin, Uri Shapira, Shucheng Yu
We study the limiting distribution of the rational points under a horizontal translation along a sequence of expanding closed horocycles on the modular surface. Using spectral meth…
On continued fraction expansions of quadratic irrationals in positive characteristic
Frédéric Paulin, Uri Shapira
Let be a prime polynomial in the variable over a finite field and let be a quadratic irrational in the field of formal Laurant series in the variable . We study…
Limiting Distributions of Translates of Divergent Diagonal Orbits
Uri Shapira, Cheng Zheng
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforward…
Equidistribution of divergent orbits of the diagonal group in the space of lattices
Ofir David, Uri Shapira
We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: The discrimin…
Dynamics on the space of 2-lattices in 3-space
Oliver Sargent, Uri Shapira
We study the dynamics of and its subgroups on the homogeneous space consisting of homothety classes of rank-2 discrete subgroups of . We focus…
Equidistribution of divergent orbits and continued fraction expansion of rationals
Ofir David, Uri Shapira
We establish an equidistribution result for push-forwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an applicat…
Dimension bound for badly approximable grids
Seonhee Lim, Nicolas de Saxcé, Uri Shapira
We show that for almost any vector in , for any there exists such that the dimension of the set of vectors satisfying $\liminf_{k\to\infty} k^{1/n…
Stable lattices and the diagonal group
Uri Shapira, Barak Weiss
Inspired by work of McMullen, we show that any orbit of the diagonal group in the space of lattices accumulates on the set of stable lattices. As consequences, we settle a conjectu…
Asymptotic metric behavior of random Cayley graphs of finite abelian groups
Uri Shapira, Reut Zuck
Using methods of Marklof and Strömbergsson we establish several limit laws for metric parameters of random Cayley graphs of finite abelian groups with respect to a randomly chosen…
Escape of mass in homogeneous dynamics in positive characteristic
Alexander Kemarsky, Frédéric Paulin, Uri Shapira
We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of 2-lattices, when varied along rays o…
Full escape of mass for the diagonal group
Uri Shapira
Building on the work of Cassels we prove the existence of infinite families of compact orbits of the diagonal group in the space of lattices which accumulate only on the divergent…
Closed orbits for the diagonal group and well-rounded lattices
Michael Levin, Uri Shapira, Barak Weiss
Curt McMullen showed that every compact orbit for the action of the diagonal group on the space of lattices contains a well-rounded lattice. We extend this to all closed orbits.
On stable lattices and the diagonal group
Uri Shapira, Barak Weiss
Inspired by work of McMullen, we show that any orbit for the action of the diagonal group on the space of lattices, accumulates on a stable lattice. We use this to settle a conject…
Equidistribution of primitive rational points on expanding horospheres
Manfred Einsiedler, Shahar Mozes, Nimish Shah +1
We confirm a conjecture of Jens Marklof regarding the equidistribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersect…
On the Mordell-Gruber spectrum
Uri Shapira, Barak Weiss
We investigate the Mordell constant of certain families of lattices, in particular, of lattices arising from totally real fields. We define the almost sure value k_mu of the Mordel…
On the evolution of continued fractions in a fixed quadratic field
Menny Aka, Uri Shapira
We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the `normal' stat…
Homogeneous orbit closures and applications
Elon Lindenstrauss, Uri Shapira
We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit closures, as well as examples of…
Grids with dense values
Uri Shapira
Given a continuous function from Euclidean space to the real line, we analyze (under some natural assumption on the function), the set of values it takes on translates of lattices.…
Diophantine approximations on fractals
Manfred Einsiedler, Lior Fishman, Uri Shapira
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any poi…
Measure theoretical entropy of covers
Uri Shapira
In this paper we introduce three notions of measure theoretical entropy of a measurable cover U in a measure theoretical dynamical system. Two of them were already introduced in [R…
On a generalization of Littlewood's conjecture
Uri Shapira
We present a class of lattices in R^d (d >= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amo…