Vol. 10, No. 6, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 8, 1793–2041
Issue 7, 1539–1791
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Fuglede's spectral set conjecture for convex polytopes

Rachel Greenfeld and Nir Lev

Vol. 10 (2017), No. 6, 1497–1538
Abstract

Let Ω be a convex polytope in d . We say that Ω is spectral if the space L2(Ω) admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that Ω is spectral if and only if it can tile the space by translations. It is known that if Ω tiles then it is spectral, but the converse was proved only in dimension d = 2, by Iosevich, Katz and Tao.

By a result due to Kolountzakis, if a convex polytope Ω d is spectral, then it must be centrally symmetric. We prove that also all the facets of Ω are centrally symmetric. These conditions are necessary for Ω to tile by translations.

We also develop an approach which allows us to prove that in dimension d = 3, any spectral convex polytope Ω indeed tiles by translations. Thus we obtain that Fuglede’s conjecture is true for convex polytopes in 3 .

Keywords
Fuglede's conjecture, spectral set, tiling, convex polytope
Mathematical Subject Classification 2010
Primary: 42B10, 52C22
Milestones
Received: 5 March 2017
Accepted: 29 May 2017
Published: 14 July 2017
Authors
Rachel Greenfeld
Department of Mathematics
Bar-Ilan University
Ramat-Gan 52900
Israel
Nir Lev
Department of Mathematics
Bar-Ilan University
Ramat-Gan 52900
Israel