20 research outputs found
Fuglede's conjecture fails in dimension 4
In this note we give an example of a set \W\subset \R^4 such that L^2(\W)
admits an orthonormal basis of exponentials \{\frac{1}{|\W |^{1/2}}e^{2\pi i
x, \xi}\}_{\xi\in\L} for some set \L\subset\R^4, but which does not tile
by translations. This improves Tao's recent 5-dimensional example, and
shows that one direction of Fuglede's conjecture fails already in dimension 4.
Some common properties of translational tiles and spectral sets are also
proved.Comment: 6 page
A Fourier analytic approach to the problem of mutually unbiased bases
We give an entirely new approach to the problem of mutually unbiased bases
(MUBs), based on a Fourier analytic technique in additive combinatorics. The
method provides a short and elegant generalization of the fact that there are
at most MUBs in \Co^d. It may also yield a proof that no complete
system of MUBs exists in some composite dimensions -- a long standing open
problem.Comment: 11 page
On the existence of flat orthogonal matrices
In this note we investigate the existence of flat orthogonal matrices, i.e.
real orthogonal matrices with all entries having absolute value close to
. Entries of correspond to
Hadamard matrices, so the question of existence of flat orthogonal matrices can
be viewed as a relaxation of the Hadamard problem.Comment: 10 page