Related papers: Intersections of diagonal orbits

In this note we study the distribution of the intersections between certain translates of closed orbits of the positive diagonal subgroup in $\mathrm{SL}(3, \mathbb{Z}) \backslash \mathrm{SL}(3, \mathbb{R})$ with a maximal parabolic…

Let G=SL(n,R) with n>5. We construct examples of lattices Gamma of G, subgroup A of the diagonal group and points x in G/Gamma such that the closure of the orbit Ax is not homogeneous but does not factors through the action of a…

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give…

Let S be an nXn real symmetric matrix with spectral decomposition S=Q^T Lambda Q, where Q is an orthogonal matrix and Lambda is diagonal with simple spectrum {lambda_1,..., lambda_n}. Also let O_S e R_S be the orbits by conjugation of S by,…

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.

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 conjecture of Ramharter concerning the asymptotic behaviour of the…

In this paper, we study the orbit intersection problem for the linear space and the algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi_1, \Phi_{2}: K^d…

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 conjecture of Ramharter about Mordell's constant, get new proofs of…

We give examples of $n \times n$ matrices $A$ and $B$ over the filed $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ such that for almost every column vector $x \in \mathbb{K}^n$, the orbit of $x$ under the action of the semigroup generated by $A$…

Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be…

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus $(\C^*)^n$ is a finite group which can be determined using the torsion part of some associated lattice. Applications are…

We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the…

Let $A\in \mathbb{R}^{N\times N}$ and $\mathrm{SO}_n:=\{ U \in \mathbb{R}^{N \times N}:UU^t=I_n,\det U>0\}$ be the set of $n\times n$ special orthogonal matrices. Define the (real) special orthogonal orbit of $A$ by \[…

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…

Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is…

We classify complex hyperplane arrangements $\mathcal A$ whose intersection posets $L(\mathcal A)$ satisfy $L(\mathcal A)=\pi_i^{-1}\circ\pi_i\bigl(L(\mathcal A)\bigr)$ for $i=1,\dots,n$. Here $\pi_i$ denotes the projection from $\mathbb…

For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.

We know that $\mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $\mathbb{Z}^m_n= \mathbb{Z}_n \times\mathbb{Z}_n\times...\times\mathbb{Z}_n$ be the Cartesian product of $m$ rings…

A paper by Cavenagh and Wanless diagnosed the possible intersection of any two transversals of the back circulant Latin square B_n, and used the result to completely determine the spectrum for 2-way k-homogeneous latin trades. We give a…

This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an "orthogonal" basis…