A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on finite Abelian groups show that Fuglede conjecture is false in high dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the finite groups ZN. Based on the Coven- Meyerowitz conjecture, it has been proved that if any spectral set in Z satisfies the the Coven-Meyerowitz properties, then every spectral set in R is a tile. We will present some of the main results related to integer tiles and give a self-contained description of the theory with detailed proofs.
| Identifer | oai:union.ndltd.org:ucf.edu/oai:stars.library.ucf.edu:etd-5709 |
| Date | 01 January 2014 |
| Creators | Li, Shasha |
| Publisher | STARS |
| Source Sets | University of Central Florida |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Electronic Theses and Dissertations |
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