Fuglede’s conjecture fails in 4 dimensions over odd prime fields

Abstract

We describe computer programs accompanying our paper [3] and show that running them suffices to verify Fuglede's conjecture in Z 5 2 and Z 6 2 .

A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

“…However, this exception seems to be due to the small number of points and is not an indication for larger dimensions. These results were found independently and with different methods by Ferguson and Sothanaphan [6].…”

Fuglede’s conjecture holds on ℤ_{𝕡}²×ℤ_{𝕢}

“…It was proved in [10] that for Z 2 p Fuglede's conjecture holds for every prime p. On the other hand, it was shown in [1] that the spectral tile direction of the conjecture does not hold for Z 5 p if p is an odd prime and for Z 4 p if p ≡ 3 (mod 4) is a prime. This was strengthened by Ferguson and Sothanaphan by exhibiting a non-spectral tile for Z 4 p , see [7]. If p = 2, then the situation is slightly different.…”

“…If p = 2, then the situation is slightly different. It was shown that Fuglede's conjecture fails for Z d 2 if d ≥ 10 [7], and holds if d ≤ 6 [7,8]. For 7 ≤ d ≤ 9 the answer is not known.…”

On the structure of spectral and tiling subsets of cyclic groups

“…We return to the setting of this paper, namely finite Abelian groups: for finite Abelian groups with two generators, it has been shown that Fuglede's conjecture holds in Z p × Z p [11], a result later extended to Z p × Z p 2 [28]. When the generators are at least four, the Spectral⇒Tiling direction fails when the cardinality of the group is odd [8].…”