Ethan Coven and Aaron Meyerowitz have a paper <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&ved=0CEsQFjAF&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.45.3075%26rep%3Drep1%26type%3Dpdf&ei=dQ3bTuWGD5DWiALhzujzDA&usg=AFQjCNHZIGYs4vDNNe8Cz8j2MJBkYx46PQ">"Tiling the integers with translates of one finite set"</a> (cited recently on <a href="http://terrytao.wordpress.com/2011/11/19/some-notes-on-the-coven-meyerowitz-conjecture/">Terry Tao's blog</a>) 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.