Recent Issues
Volume 17, 10 issues
Volume 17
Issue 10, 3371–3670
Issue 9, 2997–3369
Issue 8, 2619–2996
Issue 7, 2247–2618
Issue 6, 1871–2245
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377
Volume 16, 10 issues
Volume 16
Issue 10, 2241–2494
Issue 9, 1989–2240
Issue 8, 1745–1988
Issue 7, 1485–1744
Issue 6, 1289–1483
Issue 5, 1089–1288
Issue 4, 891–1088
Issue 3, 613–890
Issue 2, 309–612
Issue 1, 1–308
Volume 15, 8 issues
Volume 15
Issue 8, 1861–2108
Issue 7, 1617–1859
Issue 6, 1375–1616
Issue 5, 1131–1373
Issue 4, 891–1130
Issue 3, 567–890
Issue 2, 273–566
Issue 1, 1–272
Volume 14, 8 issues
Volume 14
Issue 8, 2327–2651
Issue 7, 1977–2326
Issue 6, 1671–1976
Issue 5, 1333–1669
Issue 4, 985–1332
Issue 3, 667–984
Issue 2, 323–666
Issue 1, 1–322
Volume 13, 8 issues
Volume 13
Issue 8, 2259–2480
Issue 7, 1955–2257
Issue 6, 1605–1954
Issue 5, 1269–1603
Issue 4, 945–1268
Issue 3, 627–944
Issue 2, 317–625
Issue 1, 1–316
Volume 12, 8 issues
Volume 12
Issue 8, 1891–2146
Issue 7, 1643–1890
Issue 7, 1397–1644
Issue 6, 1397–1642
Issue 5, 1149–1396
Issue 4, 867–1148
Issue 3, 605–866
Issue 2, 259–604
Issue 1, 1–258
Volume 11, 8 issues
Volume 11
Issue 8, 1841–2148
Issue 7, 1587–1839
Issue 6, 1343–1586
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261
Volume 10, 8 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 9
Issue 8, 1772–2050
Issue 7, 1523–1772
Issue 6, 1285–1522
Issue 5, 1019–1283
Issue 4, 773–1018
Issue 3, 515–772
Issue 2, 259–514
Issue 1, 1–257
Volume 8, 8 issues
Volume 8
Issue 8, 1807–2055
Issue 7, 1541–1805
Issue 6, 1289–1539
Issue 5, 1025–1288
Issue 4, 765–1023
Issue 3, 513–764
Issue 2, 257–511
Issue 1, 1–255
Volume 7, 8 issues
Volume 7
Issue 8, 1713–2027
Issue 7, 1464–1712
Issue 6, 1237–1464
Issue 5, 1027–1236
Issue 4, 771–1026
Issue 3, 529–770
Issue 2, 267–527
Issue 1, 1–266
Volume 6, 8 issues
Volume 6
Issue 8, 1793–2048
Issue 7, 1535–1791
Issue 6, 1243–1533
Issue 5, 1001–1242
Issue 4, 751–1000
Issue 3, 515–750
Issue 2, 257–514
Issue 1, 1–256
Volume 5, 5 issues
Volume 5
Issue 5, 887–1173
Issue 4, 705–885
Issue 3, 423–703
Issue 2, 219–422
Issue 1, 1–218
Volume 4, 5 issues
Volume 4
Issue 5, 639–795
Issue 4, 499–638
Issue 3, 369–497
Issue 2, 191–367
Issue 1, 1–190
Volume 3, 4 issues
Volume 3
Issue 4, 359–489
Issue 3, 227–358
Issue 2, 109–225
Issue 1, 1–108
Volume 2, 3 issues
Volume 2
Issue 3, 261–366
Issue 2, 119–259
Issue 1, 1–81
Volume 1, 3 issues
Volume 1
Issue 3, 267–379
Issue 2, 127–266
Issue 1, 1–126
Abstract
Let
Ω be a convex
polytope in
ℝ d . We
say that
Ω is spectral
if the space
L 2 ( 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
⊂ ℝ 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
ℝ 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