Corrigendum and Addendum to: Haar Bases for L^2(R^n) and Algebraic Number Theory

Gro chenig and Madych showed that a Haar-type orthonormal wavelet basis of L(R) can be constructed from the characteristic function /Q of a set Q if and only if Q is an affine image of an integral self-affine tile T which tiles R using the integer lattice Z. An integral self-affine tile T=T(A, D) is the attractor of an iterated function system T= i=1 A (T+di) where A # Mn(Z) is an expanding n_n integer matrix and the digit set D=[d1 , d2 , ..., dm] Z has m=|det(A)|, provided that the Lebesgue measure +(T )>0. Two necessary conditions for T(A, D) to tile R with the integer lattice Z are that D be a complete set of coset representatives of Z A(Z) and that Z[A, D]=Z, where Z[A, D] is the smallest A-invariant lattice containing all [di&dj : i{ j]. These two conditions are necessary and sufficient in the special case that |det(A)|=2. We study these two conditions for an arbitrary matrix A # Mn(Z). We prove that a digit set D satisfying the two conditions exists whenever |det(A)| n+1. When |det(A)|=2 there are number-theoretic obstructions to the existence of such D. Using these we exhibit a (non-expanding) A # M2(Z) for which no digit set has Z[A, D]=Z. However we show that for all expanding integer matrices A in dimensions 2 and 3, there exists some digit set D that satisfies the two conditions. Could this be true for all expanding integer matrices in dimensions n 4? A necessary condition is that the (non-Galois) field Q( n 2) have class number one for all n 4. 1996 Academic Press, Inc.