Bornivorous set | EPFL Graph Search

In functional analysis, a subset of a real or complex vector space that has an associated vector bornology is called bornivorous and a bornivore if it absorbs every element of If is a topological vector space (TVS) then a subset of is bornivorous if it is bornivorous with respect to the von-Neumann bornology of . Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces. If is a TVS then a subset of is called and a if absorbs every bounded subset of An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets). A linear map between two TVSs is called if it maps Banach disks to bounded disks. A disk in is called if it absorbs every Banach disk. An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded. A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is ""). Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin. Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores. Suppose is a vector subspace of finite codimension in a locally convex space and If is a barrel (resp. bornivorous barrel, bornivorous disk) in then there exists a barrel (resp. bornivorous barrel, bornivorous disk) in such that Every neighborhood of the origin in a TVS is bornivorous. The convex hull, closed convex hull, and balanced hull of a bornivorous set is again bornivorous. The preimage of a bornivore under a bounded linear map is a bornivore. If is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore. Let be as a vector space over the reals. If is the balanced hull of the closed line segment between and then is not bornivorous but the convex hull of is bornivorous.

Moments of the number of points in a bounded set for number field lattices

Maryna Viazovska, Nihar Prakash Gargava, Vlad Serban

We examine the moments of the number of lattice points in a fixed ball of volume

V

for lattices in Euclidean space which are modules over the ring of integers of a number field

K

. In particular, denoting by

ω_K

the number of roots of unity in

K

, we ...

arXiv2023