Let A \in Mn ({\Bbb Z}) be an expanding matrix with | {\det (A)} | = q and let K = {k1 \cdots kq} \subseteq {\Bbb R}n be a digit set. The set \cal T =:\cal T(A,K) = {\sumi=1\infty A-i kji : kji \in K} \subset {\Bbb R}n is called a {\it self-affine tile} if the Lebesgue measure of \cal T is positive. In this note, we consider dilation equations of the form f(x) = \sumj=1q cj f(Ax- kj) with q=\sumj=1q {cj}, cj\in {\Bbb R}, and prove that this equation has a nontrivial Lp solution (1\leq p \leq \infty) if and only if cj=1 \forall j\in {1,...,q} and \cal T is a tile.

Let A \in Mn ({\Bbb Z}) be an expanding matrix with | {\det (A)} | = q and let K = {k1 \cdots kq} \subseteq {\Bbb R}n be a digit set. The set \cal T =:\cal T(A,K) = {\sumi=1\infty A-i kji : kji \in K} \subset {\Bbb R}n is called a {\it self-affine tile} if the Lebesgue measure of \cal T is positive. In this note, we consider dilation equations of the form f(x) = \sumj=1q cj f(Ax- kj) with q=\sumj=1q {cj}, cj\in {\Bbb R}, and prove that this equation has a nontrivial Lp solution (1\leq p \leq \infty) if and only if cj=1 \forall j\in {1,...,q} and \cal T is a tile.