TY - JOUR AB - We establish effective counting results for lattice points in families of domains in real, complex and quaternionic hyperbolic spaces of any dimension. The domains we focus on are defined as product sets with respect to an Iwasawa decomposition. Several natural diophantine problems can be reduced to counting lattice points in such domains. These include equidistribution of the ratio of the length of the shortest solution (x,y) to the gcd equation bx−ay=1 relative to the length of (a,b), where (a,b) ranges over primitive vectors in a disc whose radius increases, the natural analog of this problem in imaginary quadratic number fields, as well as equidistribution of integral solutions to the diophantine equation defined by an integral Lorentz form in three or more variables. We establish an effective rate of convergence for these equidistribution problems, depending on the size of the spectral gap associated with a suitable lattice subgroup in the isometry group of the relevant hyperbolic space. The main result underlying our discussion amounts to establishing effective joint equidistribution for the horospherical component and the radial component in the Iwasawa decomposition of lattice elements. AU - Horesh, Tal AU - Nevo, Amos ID - 14245 IS - 2 JF - Pacific Journal of Mathematics SN - 0030-8730 TI - Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution VL - 324 ER - TY - JOUR AB - We count primitive lattices of rank d inside Zn as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subspace that a lattice spans, namely its projection to the Grassmannian; its homothety class and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude the joint equidistribution of these parameters. In addition to the primitive d-lattices Λ themselves, we also consider their orthogonal complements in Zn⁠, A1⁠, and show that the equidistribution occurs jointly for Λ and A1⁠. Finally, our asymptotic formulas for the number of primitive lattices include an explicit bound on the error term. AU - Horesh, Tal AU - Karasik, Yakov ID - 14717 IS - 4 JF - Quarterly Journal of Mathematics SN - 0033-5606 TI - Equidistribution of primitive lattices in ℝn VL - 74 ER - TY - JOUR AB - We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano varieties. AU - Browning, Timothy D AU - Horesh, Tal AU - Wilsch, Florian Alexander ID - 9199 IS - 10 JF - Algebra & Number Theory SN - 1937-0652 TI - Equidistribution and freeness on Grassmannians VL - 16 ER - TY - JOUR AB - Given a place ω of a global function field K over a finite field, with associated affine function ring Rω and completion Kω , the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points (a,b)∈Rω2 in the plane Kω2 , and for renormalized solutions to the gcd equation ax+by=1 . The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in \ZZ2 . AU - Horesh, Tal AU - Paulin, Frédéric ID - 12684 IS - 3 JF - Journal de Theorie des Nombres de Bordeaux SN - 1246-7405 TI - Effective equidistribution of lattice points in positive characteristic VL - 34 ER -