68 research outputs found

### Bounds for solid angles of lattices of rank three

We find sharp absolute constants $C_1$ and $C_2$ with the following property:
every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis
so that the solid angle spanned by these basis vectors lies in the interval
$[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger
class of lattices than just well-rounded, and the upper bound holds for all. We
state a technical condition on the lattice that may prevent it from satisfying
the absolute lower bound on the solid angle, in which case we derive a lower
bound in terms of the ratios of successive minima of the lattice. We use this
result to show that among all spherical triangles on the unit sphere in
$\mathbb R^N$ with vertices on the minimal vectors of a lattice, the smallest
possible area is achieved by a configuration of minimal vectors of the
(normalized) face centered cubic lattice in $\mathbb R^3$. Such spherical
configurations come up in connection with the kissing number problem.Comment: 12 pages; to appear in the Journal of Combinatorial Theory

### Frobenius problem and the covering radius of a lattice

Let $N \geq2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers.
Frobenius number of this $N$-tuple is defined to be the largest positive
integer that cannot be expressed as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$
are non-negative integers. The condition that $gcd(a_1,...,a_N)=1$ implies that
such number exists. The general problem of determining the Frobenius number
given $N$ and $a_1,...,a_N$ is NP-hard, but there has been a number of
different bounds on the Frobenius number produced by various authors. We use
techniques from the geometry of numbers to produce a new bound, relating
Frobenius number to the covering radius of the null-lattice of this $N$-tuple.
Our bound is particularly interesting in the case when this lattice has equal
successive minima, which, as we prove, happens infinitely often.Comment: 12 pages; minor revisions; to appear in Discrete and Computational
Geometr

### Hecke operators on rational functions

We define Hecke operators U_m that sift out every m-th Taylor series
coefficient of a rational function in one variable, defined over the reals. We
prove several structure theorems concerning the eigenfunctions of these Hecke
operators, including the pleasing fact that the point spectrum of the operator
U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the
simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet
characters mod L, giving rise to the result that any arithmetic function of m
that is completely multiplicative and also satisfies a linear recurrence must
be a Dirichlet character times a power of m. We also define the notions of
level and weight for rational eigenfunctions, by analogy with modular forms,
and we show the existence of some interesting finite-dimensional subspaces of
rational eigenfunctions (of fixed weight and level), whose union gives all of
the rational functions whose coefficients are quasi-polynomials.Comment: 35 pages, LaTe

### The integer point transform as a complete invariant

The integer point transform $\sigma_P$ is an important invariant of a
rational polytope $P$, and here we prove that it is a complete invariant. We
prove that it is only necessary to evaluate $\sigma_P$ at one algebraic point
in order to uniquely determine $P$. Similarly, we prove that it is only
necessary to evaluate the continuous Fourier transform of a rational polytope
$P$ at a single algebraic point, in order to uniquely determine $P$.
By relating the integer point transform to finite Fourier transforms, we show
that a finite number of integer point evaluations of $\sigma_P$ suffice in
order to uniquely determine $P$. In addition, we give an equivalent condition
for central symmetry in terms of the integer point transform, and some facts
about its local maxima. We prove many of these results for arbitrary finite
sets of integer points in $\mathbb R^d$.Comment: 12 pages, 3 figure

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