131 research outputs found
Constructions of complex Hadamard matrices via tiling Abelian groups
Applications in quantum information theory and quantum tomography have raised
current interest in complex Hadamard matrices. In this note we investigate the
connection between tiling Abelian groups and constructions of complex Hadamard
matrices. First, we recover a recent very general construction of complex
Hadamard matrices due to Dita via a natural tiling construction. Then we find
some necessary conditions for any given complex Hadamard matrix to be
equivalent to a Dita-type matrix. Finally, using another tiling construction,
due to Szabo, we arrive at new parametric families of complex Hadamard matrices
of order 8, 12 and 16, and we use our necessary conditions to prove that these
families do not arise with Dita's construction. These new families complement
the recent catalogue of complex Hadamard matrices of small order.Comment: 15 page
On the uniform distribution in residue classes of dense sets of integers with distinct sums
A set is of type if all sums ,
with , , are distinct. It is well known that the
largest such set is of size asymptotic to . For a set
of this size we show that, under mild assumptions on the size of the modulus
and on the difference N^{1/2}-\Abs{{\cal A}} (these quantities should not
be too large) the elements of are uniformly distributed in the
residue classes mod . Quantitative estimates on how uniform the distribution
is are also provided. This generalizes recent results of Lindstr\"om whose
approach was combinatorial. Our main tool is an upper bound on the minimum of a
cosine sum of terms, , all of whose positive
integer frequencies are at most in size.Comment: 5 pages, no figure
Filling a box with translates of two bricks
We give a new proof of the following interesting fact recently proved by
Bower and Michael: if a d-dimensional rectangular box can be tiled using
translates of two types of rectangular bricks, then it can also be tiled in the
following way. We can cut the box across one of its sides into two boxes, one
of which can be tiled with the first brick only and the other one with the
second brick. Our proof relies on the Fourier Transform. We also show that no
such result is true for three, or more, types of bricks
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