143 research outputs found
On similarity classes of well-rounded sublattices of
A lattice is called well-rounded if its minimal vectors span the
corresponding Euclidean space. In this paper we study the similarity classes of
well-rounded sublattices of . We relate the set of all such
similarity classes to a subset of primitive Pythagorean triples, and prove that
it has structure of a noncommutative infinitely generated monoid. We discuss
the structure of a given similarity class, and define a zeta function
corresponding to each similarity class. We relate it to Dedekind zeta of
, and investigate the growth of some related Dirichlet series,
which reflect on the distribution of well-rounded lattices. Finally, we
construct a sequence of similarity classes of well-rounded sublattices of
, which gives good circle packing density and converges to the
hexagonal lattice as fast as possible with respect to a natural metric we
define.Comment: 27 pages, 2 figures; added a lemma on Diophantine approximation by
quotients of Pythagorean triples; final version to be published in Journal of
Number Theor
Integral orthogonal bases of small height for real polynomial spaces
Let be the space of all real polynomials in variables with the
usual inner product on it, given by integrating over the unit sphere. We
start by deriving an explicit combinatorial formula for the bilinear form
representing this inner product on the space of coefficient vectors of all
polynomials in of degree . We exhibit two applications of this
formula. First, given a finite dimensional subspace of defined
over , we prove the existence of an orthogonal basis for ,
consisting of polynomials of small height with integer coefficients, providing
an explicit bound on the height; this can be viewed as a version of Siegel's
lemma for real polynomial inner product spaces. Secondly, we derive a criterion
for a finite set of points on the unit sphere in to be a spherical
-design.Comment: 10 pages; to appear in the Online Journal of Analytic Combinatoric
Heights and quadratic forms: on Cassels' theorem and its generalizations
In this survey paper, we discuss the classical Cassels' theorem on existence
of small-height zeros of quadratic forms over Q and its many extensions, to
different fields and rings, as well as to more general situations, such as
existence of totally isotropic small-height subspaces. We also discuss related
recent results on effective structural theorems for quadratic spaces, as well
as Cassels'-type theorems for small-height zeros of quadratic forms with
additional conditions. We conclude with a selection of open problems.Comment: 16 pages; to appear in the proceedings of the BIRS workshop on
"Diophantine methods, lattices, and arithmetic theory of quadratic forms", to
be published in the AMS Contemporary Mathematics serie
Siegel's lemma with additional conditions
Let be a number field, and let be a subspace of , .
Let be subspaces of of dimension less than dimension of
. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a
point in terms of heights of and . Our main tool is a counting
estimate we prove for the number of points of a subspace of inside of an
adelic cube. As corollaries to our main result we derive an explicit bound on
the height of a non-vanishing point for a decomposable form and an effective
subspace extension lemma.Comment: 12 pages, revised version, to appear in Journal of Number Theor
On effective Witt decomposition and Cartan-Dieudonne theorem
Let be a number field, and let be a symmetric bilinear form in
variables over . Let be a subspace of . A classical theorem of Witt
states that the bilinear space can be decomposed into an orthogonal sum
of hyperbolic planes, singular, and anisotropic components. We prove the
existence of such a decomposition of small height, where all bounds on height
are explicit in terms of heights of and . We also prove a special
version of Siegel's Lemma for a bilinear space, which provides a small-height
orthogonal decomposition into one-dimensional subspaces. Finally, we prove an
effective version of Cartan-Dieudonn{\'e} theorem. Namely, we show that every
isometry of a regular bilinear space can be represented as a
product of reflections of small heights with an explicit bound on heights in
terms of heights of , , and .Comment: 16 pages, revised and corrected version, to appear in Canadian
Journal of Mathematic
Revisiting the hexagonal lattice: on optimal lattice circle packing
In this note we give a simple proof of the classical fact that the hexagonal
lattice gives the highest density circle packing among all lattices in .
With the benefit of hindsight, we show that the problem can be restricted to
the important class of well-rounded lattices, on which the density function
takes a particularly simple form. Our proof emphasizes the role of well-rounded
lattices for discrete optimization problems.Comment: 8 pages, 1 figure; to appear in Elemente der Mathemati
Integral points of small height outside of a hypersurface
Let be a non-zero polynomial with integer coefficients in variables
of degree . We prove the existence of an integral point of small height at
which does not vanish. Our basic bound depends on and only. We
separately investigate the case when is decomposable into a product of
linear forms, and provide a more sophisticated bound. We also relate this
problem to a certain extension of Siegel's Lemma as well as to Faltings'
version of it. Finally we exhibit an application of our results to a discrete
version of the Tarski plank problem.Comment: 16 pages, revised version, to appear in Monatshefte f\"{u}r
Mathemati
Small Zeros of Quadratic Forms with Linear Conditions
Given a quadratic form and linear forms in variables with
coefficients in a number field , suppose that there exists a point in
at which the quadratic form vanishes and all the linear forms do not.
Then we show that there exists a point like this of relatively small height.
This generalizes a result of D.W. Masser (1998). As a corollary of this result,
we prove an extension of Cassels' theorem on small zeros of quadratic forms
(1955) to non-singular small zeros over a number field.Comment: 11 pages, to appear in Journal of Number Theor
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