95 research outputs found
Riesz bases of exponentials on multiband spectra
Let be the union of finitely many disjoint intervals on the real line.
Suppose that there are two real numbers such that the length of
each interval belongs to . We use quasicrystals to
construct a discrete set of real frequencies such that the corresponding system
of exponentials is a Riesz basis in the space .Comment: 5 pages, to appear in Proceedings of the American Mathematical
Societ
Sets of bounded discrepancy for multi-dimensional irrational rotation
We study bounded remainder sets with respect to an irrational rotation of the
-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who
characterized the intervals with bounded remainder in dimension one.
First we extend to several dimensions the Hecke-Ostrowski result by
constructing a class of -dimensional parallelepipeds of bounded remainder.
Then we characterize the Riemann measurable bounded remainder sets in terms of
"equidecomposability" to such a parallelepiped. By constructing invariants with
respect to this equidecomposition, we derive explicit conditions for a polytope
to be a bounded remainder set. In particular this yields a characterization of
the convex bounded remainder polygons in two dimensions. The approach is used
to obtain several other results as well.Comment: To appear in Geometric And Functional Analysi
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