MULTIPLE LATTICE TILES AND RIESZ BASES OF EXPONENTIALS

作者：Kolountzakis Mihail N^*

来源：Proceedings of the American Mathematical Society, 2015, 143(2): 741-747.

• Riesz bases of exponentials
• tiling

Suppose Omega subset of R-d is a bounded and measurable set and Lambda subset of R-d is a lattice. Suppose also that Omega tiles multiply, at level k, when translated at the locations Lambda. This means that the.-translates of Omega cover almost every point of R-d exactly k times. We show here that there is a set of exponentials exp(2 pi it center dot x), t is an element of T, where T is some countable subset of R-d, which forms a Riesz basis of L-2(Omega). This result was recently proved by Grepstad and Lev under the extra assumption that Omega has boundary of measure 0, using methods from the theory of quasicrystals. Our approach is rather more elementary and is based almost entirely on linear algebra. The set of frequencies T turns out to be a finite union of shifted copies of the dual lattice Lambda*. It can be chosen knowing only. and k and is the same for all Omega that tile multiply with..

• 出版日期2015-2