Characterization of self-affine tile digit sets on R n *

Let R be an n × n expanding matrix with integer entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set D ⊂ Z n so that the integral self-affine set T ( R , D ) is a translational tile on R n . In this paper, we introduce a notion of skew-product-form digit set which is a very general class of tile digit sets. We prove that under certain conditions, T ( R , D ) is a self-affine tile if and only if there exists m ⩾ 1 such that D m = D + R D + ⋯ + R m − 1 D is a skew-product-form digit set. Especially, we show that in the one-dimensional case, T ( b , D ) is a self-similar tile if and only if there exists m ⩾ 1 such that D m is a skew-product-form digit set. Notice that T ( b , D ) = T ( b m , D m ) . In this sense, we fully characterize the self-similar tile digit sets in R 1 . As an application, we establish that all self-similar tiles T ( b , D ) are spectral sets when A and B j ( 0 ⩽ j ⩽ s ) in the corresponding skew-product-form digit set satisfy ( T 1 ) and ( T 2 ) conditions.