Ethan Coven and Aaron Meyerovitch have a paper "Tiling the integers with translates of one finite set" (cited recently on Terry Tao's blog) about tiling the integers with a single prototile.

That is, they are looking for conditions under which subsets of the integers, $A$ say, such that there are $n_1,n_2,n_3,...$ such that $n_1+A$, $n_2+A$, $\ldots$ disjointly cover all of the integers.

In Coven and Meyerovitch's paper, two conditions are given directly in terms of cyclotomic polynomials for $A$ to have this property. If both conditions are satisfied, then $A$ tiles the integers. On the other hand, if $A$ tiles the integers, then one of the conditions is shown to be satisfied. They conjecture that the second condition must also be satisfied.

Ethan Coven and Aaron Meyerowitz have a paper "Tiling the integers with translates of one finite set" (cited recently on Terry Tao's blog) about tiling the integers with a single prototile.

That is, they are looking for conditions under which subsets of the integers, $A$ say, such that there are $n_1,n_2,n_3,...$ such that $n_1+A$, $n_2+A$, $\ldots$ disjointly cover all of the integers.

In Coven and Meyerowitz's paper, two conditions are given directly in terms of cyclotomic polynomials for $A$ to have this property. If both conditions are satisfied, then $A$ tiles the integers. On the other hand, if $A$ tiles the integers, then one of the conditions is shown to be satisfied. They conjecture that the second condition must also be satisfied.