Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

This condition is necessary, but not sufficient. Though rather obvious, it was apparently not spotted before 1998.

3.1.4 Condition $(T 2)$

While condition $(T1)$ explains what happens to » visible « cyclotomic factors, condition $(T 2)$ tells where some of the » invisible « cyclotomic factors (those factors with value 1 in 0) must lie:

$(T2)$: If $pa,qb,...,rg (- SA$ are powers of different primes $p,q,...,r$ , then $Ppaqb...rg$ is a divisor of $A(x)$ .

This is a kind of stability property for the set $D(A) = {d,Pd |A(x)}$ . So we have several factors of $A(x)$ :

The $Ppa$ , subject to condition $(T1)$ .
The $P a b g p q ...r$ with $pa,qb,...rg (- S A$ : this is related to condition $(T2)$ .
Compound cyclotomic factors $Pa.b...$ with $a (- SA,b (- SB$ ,which are factors of either $A(x)$ or $B(x)$ .
Non cyclotomic factors (scarce).

Recently Coven and Meyerowitz(Coven and Meyerowitz, 1999) proved the following rather easy theorem:

Theorem 11 If $A$ satisfies $(T 1)$ and $(T2)$ , then $A$ tiles.

The proof uses only elementary facts about cyclotomic polynomials and their products. They also prove a more difficult theorem, using essentially the important Theorem 4:

Theorem 12 If $A$ tiles and $|A |$ has only two prime factors, then $A$ satisfies $(T1)$ and $(T2)$ .

So $(T1)+ (T2)$ is sufficient for $A$ to tile in any case, and also necessary for »simple« values of $|A|$ . Maybe this a necessary and sufficient condition for any $|A |$ , with any number of prime factors: no counter example is known yet.

This would be a substantial part of a proof of the spectral hypothesis for (finite) tilings of $Z$ , according to (Laba, 2002).

3.1.5 An Example and an Answer

To understand the conditions $(T 1)$ and $(T2)$ above, let us look at a random example, given by Vuza’s algorithm (Vuza, 1990-91) for constructing aperiodic canons:

A = {10,18,26,28,36,44} B = {12,18,19,23,24,43,47,48,54,60,67,71}.