17 research outputs found
A simple proof of a reverse Minkowski theorem for integral lattices
We prove that for any integral lattice
(that is, a lattice such that the inner product is an integer for all ) and any positive integer ,
giving a nearly tight reverse Minkowski theorem for integral lattices
Kissing numbers and transference theorems from generalized tail bounds
We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a
lattice to a wide class of test functions. From this we obtain quite general
transference bounds, as well as bounds on the number of lattice points
contained in certain bodies. As applications, we bound the lattice kissing
number in norms by for , and also give
a proof of a new transference bound in the norm.Comment: Previous title: "Generalizations of Banaszczyk's transference
theorems and tail bound
Universal quadratic forms and Dedekind zeta functions
We study universal quadratic forms over totally real number fields using
Dedekind zeta functions. In particular, we prove an explicit upper bound for
the rank of universal quadratic forms over a given number field , under the
assumption that the codifferent of is generated by a totally positive
element. Motivated by a possible path to remove that assumption, we also
investigate the smallest number of generators for the positive part of ideals
in totally real numbers fields.Comment: 12 pages. Preprin
On lattice extensions
A lattice is said to be an extension of a sublattice of smaller
rank if is equal to the intersection of with the subspace spanned
by . The goal of this paper is to initiate a systematic study of the
geometry of lattice extensions. We start by proving the existence of a
small-determinant extension of a given lattice, and then look at successive
minima and covering radius. To this end, we investigate extensions (within an
ambient lattice) preserving the successive minima of the given lattice, as well
as extensions preserving the covering radius. We also exhibit some interesting
arithmetic properties of deep holes of planar lattices.Comment: 18 pages, 4 figures; to appear in Monatshefte f\"ur Mathemati