Algorithms to tile the infinite grid with finite clusters

Abstract

We say that a subset T of Z/sup 2/, the two dimensional infinite grid, tiles Z/sup 2/ if we can cover Z/sup 2/ with non-overlapping translates of T. No algorithm is known to decide whether a finite T/spl sube/Z/sup 2/ tiles Z/sup 2/. Here we present two algorithms, one for the case when |T| is prime, and another for the case when |T|=4. Both algorithms generalize to the case, where we replace Z/sup 2/ with all arbitrary finitely generated Abelian group. As a by-product of our results we partially settle the Periodic Tiling Conjecture raised by J. Lagarias and Y. Wang (1997), and we also get the following generalization of a theorem of L.Redei (1965): Let G be a (finite or infinite) Abelian group G with a generator set T of prime cardinality such, that 0/spl isin/T, and there is a set T'/spl sube/G with the property that for every g/spl isin/G there are unique t/spl isin/T, t'/spl isin/T' such that g=t+t'. Then T' can be replaced with a subgroup of G, that also has the above property.