20 research outputs found
A discrete analogue for Minkowski's second theorem on successive minima
The main result of this paper is an inequality relating the lattice point
enumerator of a 3-dimensional, 0-symmetric convex body and its successive
minima. This is an example of generalization of Minkowski's theorems on
successive minima, where the volume is replaced by the discrete analogue, the
lattice point enumerator. This problem is still open in higher dimensions,
however, we introduce a stronger conjecture that shows a possibility of proof
by induction on the dimension.Comment: 19 page
A note on Gabor frames in finite dimensions
The purpose of this note is to present a proof of the existence of Gabor
frames in general linear position in all finite dimensions. The tools developed
in this note are also helpful towards an explicit construction of such a frame,
which is carried out in the last section. This result has applications in
signal recovery through erasure channels, operator identification, and
time-frequency analysis.Comment: 10 page
A linear programming approach to Fuglede's conjecture in
We present an approach to Fuglede's conjecture in using
linear programming bounds, obtaining the following partial result: if
with , then is
not spectral.Comment: 13 page
On the structure of spectral and tiling subsets of cyclic groups
The purpose of this paper is to investigate the properties of spectral and
tiling subsets of cyclic groups, with an eye towards the spectral set
conjecture in one dimension, which states that a bounded measurable subset of
accepts an orthogonal basis of exponentials if and only if it
tiles by translations. This conjecture is strongly connected to
its discrete counterpart, namely that in every finite cyclic group, a subset is
spectral if and only if it is a tile. The tools presented herein are
refinements of recent ones used in the setting of cyclic groups; the structure
of vanishing sums of roots of unity is a prevalent notion throughout the text,
as well as the structure of tiling subsets of integers. We manage to prove the
conjecture for cyclic groups of order , when one of the exponents is
or when , and also prove that a tiling subset of a cyclic
group of order is spectral.Comment: 35 pages; removed one incorrect reference from version
Polyhedral Gauss Sums, and polytopes with symmetry
We define certain natural finite sums of 'th roots of unity, called
, that are associated to each convex integer polytope , and which
generalize the classical -dimensional Gauss sum defined over , to higher dimensional abelian groups and integer polytopes.
We consider the finite Weyl group , generated by the reflections
with respect to the coordinate hyperplanes, as well as all permutations of the
coordinates; further, we let be the group generated by
as well as all integer translations in . We prove
that if multi-tiles under the action of , then we
have the closed form . Conversely, we also prove
that if is a lattice tetrahedron in , of volume , such
that , for , then there is
an element in such that is the fundamental tetrahedron
with vertices , , , .Comment: 18 pages, 2 figure
Computing the covering radius of a polytope with an application to lonely runners
We are concerned with the computational problem of determining the covering
radius of a rational polytope. This parameter is defined as the minimal
dilation factor that is needed for the lattice translates of the
correspondingly dilated polytope to cover the whole space. As our main result,
we describe a new algorithm for this problem, which is simpler, more efficient
and easier to implement than the only prior algorithm of Kannan (1992).
Motivated by a variant of the famous Lonely Runner Conjecture, we use its
geometric interpretation in terms of covering radii of zonotopes, and apply our
algorithm to prove the first open case of three runners with individual
starting points.Comment: 22 pages, 4 tables, 2 figures, revised versio