Related papers: Asymptotics for pseudo-Anosov elements in Teichmul…
A lattice is called well-rounded, if its lattice vectors of minimal length span the ambient space. We show that there are interesting connections between the existence of well-rounded sublattices and coincidence site lattices (CSLs).…
This paper is concerned with discrete, uniform subgroups (lattices) of oscillator groups, which are certain semidirect products of the Heisenberg group and the additive group of real numbers. The present paper rectifies the uncertainties in…
We exhibit Anosov subgroups of $\mathsf{SL}_d(\mathbb{R})$ that do not embed discretely in any rank-$1$ simple Lie group of noncompact type, or indeed, in any finite product of such Lie groups. These subgroups are isomorphic to free…
We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by applying round-off to planar rotations. All orbits of these maps are conjectured to be periodic. We let the angle of rotation approach pi/2, and…
Let $g\colon L\rightarrow L$ be an atoroidal, endperiodic map on an infinite type surface $L$ with no boundary and finitely many ends, each of which is accumulated by genus. By work of Landry, Minsky, and Taylor, $g$ is isotopic to a spun…
We investigate the asymptotic behavior of the distribution of primitive lattice points in a symmetric Borel set $S_d\subset\mathbb R^d$ as $d$ goes to infinity, under certain volume conditions on $S_d$. Our main technique involves exploring…
We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion…
The Gr\"atzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable…
Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain ${\mathsf M}_3$-sublattices). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are $u_l, u_r \in [o, i] - \{o,i\}$ such that $o = u_l…
We propose and analyze a class of particle methods for the Vlasov equation with a strong external magnetic field in a torus configuration. In this regime, the time step can be subject to stability constraints related to the smallness of…
We prove the exponent $4/3$ for the lattice point discrepancy of a torus in $\mathbb{R}^3$ (generated by the rotation of a circle around the $z$ axis). The exponent comes from a diagonal term and it seems a natural limit for any approach…
Two semigroups are lattice isomorphic if the lattices of their subsemigroups are isomorphic, and a class of semigroups is lattice closed if it contains every semigroup which is lattice isomorphic to some semigroup from that class. An…
We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of…
The aim of this article is to extend results of M.~Popov and second named author about orthogonally additive narrow operators on vector lattices. The main object of our investigations are an orthogonally additive narrow operators between…
We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient. As a…
We show that the number of lattice points lying in a thin annulus has a Gaussian value distribution if the width of the annulus tends to zero sufficiently slowly as we increase the inner radius.
The Teichm\"uller space $\mathcal{T}(\Sigma)$ of a surface $\Sigma$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(\Sigma)$. We give the asymptotic behavior of the lengths…
The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$…
We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending…
A discrete set $A$ in the Euclidian space is almost periodic if the measure with the unite masses at points of the set is almost periodic in the weak sense. We investigate properties of such sets in the case when $A-A$ is discrete. In…