相关论文: Horospheres and Farey fractions

This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R^p as the number of points n -> infinity, while the dimension p is either fixed or growing with n. For both…

We consider rational points on the sphere and investigate their equidistribution in shrinking spherical caps. For the two-dimensional sphere, we leverage Hecke operators to obtain a significantly improved small-scale equidistribution bound,…

In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of values of numerators of differences of consecutive elements in this…

Hard-sphere colloids are popular as models for testing fundamental theories in condensed matter and statistical physics, from crystal nucleation to the glass transition. A single parameter, the volume fraction (phi), characterizes an ideal,…

We study the asymptotic stability for large times of homogeneous stationary states for the nonlinear Hartree equation for density matrices in Rd for d\geq3. We can reach both the optimal Sobolev and Schatten exponents for the initial data,…

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space $\mathbb{R}^d\backslash\{0\}$. We construct simple…

We consider the homogenization of parabolic equations with large spatially-dependent potentials modeled as Gaussian random fields. We derive the homogenized equations in the limit of vanishing correlation length of the random potential. We…

We study two types of problems for polynomial Farey fractions. For a positive integer $Q$, and polynomial $P(x)\in\mathbb{Z}[X]$ with $P(0)=0$, we define polynomial Farey fractions as \[\mathcal{F}_{Q,P}:=\left\{\frac{a}{q}: 1\leq a\leq…

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are the spheres in the Euclidean spaces, the real, complex and quaternionic…

This work focuses on an improved fractional Sobolev inequality with a remainder term involving the Hardy-Littlewood-Sobolev inequality which has been proved recently. By extending a recent result on the standard Laplacian to the fractional…

We model discrete spatial solitons in a periodic nonlinear medium encompassing any degree of transverse non locality. Making a convenient reference to a widely used material -nematic liquid crystals-, we derive a new form of the discrete…

Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…

In this PhD Thesis we investigate the geometry of random fields on compact Riemannian manifolds, in particular the two-dimensional sphere. In the first part, we characterize isotropic Gaussian fields on homogeneous spaces of a compact group…

Let G be a semisimple Lie group without compact factor and $\Gamma$ < G a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on tori. We prove that these periodic tori…

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random…

We confirm a conjecture of Jens Marklof regarding the equidistribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersecting the expanded horosphere with a certain manifold of…

I critically discuss in a pedagogical and phenomenological way a few crucial tests challenging the claims by Pietronero and collaborators that there is no evidence from available galaxy catalogues that the Universe is actually homogeneous…

Over the past two decades or so, it has become increasingly appparent that, out to quite large distances, galaxies are distributed in a quasi-fractal fashion with fractal dimension $D \approx 2$. Whether or not this behaviour continues onto…

An overview of some recent developments in inhomogeneous models is presented. As the volume and precision of cosmological data improves, it will become more and more essential to understand the non-linear behaviour of the Einstein field…