Related papers: Lattice Delone simplices with super-exponential vo…
We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves…
By studying the volume of a generalized difference body, this paper presents the first nontrivial lower bound for the lattice covering density by $n$-dimensional simplices.
Let $\mathcal{T}^d(1)$ be the set of all $d$-dimensional simplices $T$ in $\real^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $\mathcal{T}^d(1)$ is finite up to affine…
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a…
Let I be a dense linear order with a left endpoint but no right endpoint. We consider the lattice L(I) of finite unions of closed intervals of I. This lattice arises naturally in the setting of o-minimality, as these are precisely the…
We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly…
We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions…
We show that all stack-sorting polytopes are simplices. Furthermore, we show that the stack-sorting polytopes generated from $Ln1$ permutations have relative volume 1. We establish an upper bound for the number of lattice points in a…
We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.
A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…
We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit uncountable family of (transcendental) real numbers that are not realised as the simplicial volume of a…
We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.
We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions.…
By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volume of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a…
We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive…
We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/2^n, which matches, up to a multiplicative constant, the best known density of a…
We consider the problem of finding lower bounds on the number of unlabeled $n$-element lattices in some lattice family. We show that if the family is closed under vertical sum, exponential lower bounds can be obtained from vertical sums of…
A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the…
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…