Characterization of tile digit sets with prime determinants

Abstract

For an expanding integral s× s matrix A with |det A|= p, it is well known that if D={d 0,…,d p−1}⊂ Z s is a complete set of coset representatives of Z s/A Z s , then T(A, D) is a self-affine tile. In this paper we show that if p is a prime, such D actually characterizes the tile digit sets provided that span( D)= R s . This result is known for s=1, the one-dimensional case [R. Kenyon, in: Contemp. Math., vol. 135, 1992, pp. 239–264] and the question for s>1 has been considered by Lagarias and Wang [J. London Math. Soc. 53 (1996) 21–49] under some other conditions. The proof here involves a new setup to study the zeros of the mask m( ξ)= p −1∑ j=0 p−1 e 2 πi〈 ξ, d j 〉 . It can also be generalized to consider the existence of a compactly supported L 1-solution of the refinement equation (scaling function) with positive coefficients.