Haar bases for L²(ℝⁿ) and algebraic number theory

Gröchenig and Madych showed that a Haar-type orthonormal wavelet basis of L²(ℝⁿ) can be constructed from the characteristic function χ_Q of a set Q if and only if Q is an affine image of an integral self-affine tile T which tiles ℝⁿ using the integer lattice ℤⁿ. An integral self-affine tile T = T(A, script D) is the attractor of an iterated function system T = ∪^m_i=1 A-1(T+di) where A∈Μ_n(ℤ) is an expanding n×n integer matrix and the digit set script D = {d₁, d₂, ..., d_m} ⊆ℤⁿ has m = |det(A)|, provided that the Lebesgue measure μ(T)>0. Two necessary conditions for T(A, script D) to tile ℝⁿ with the integer lattice ℤⁿ are that script D be a complete set of coset representatives of ℤⁿ/A(ℤⁿ) and that ℤ[A, script D] = ℤⁿ, where ℤ[A, script 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∈Μ_n(ℤ). We prove that a digit set script 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 script D. Using these we exhibit a (non-expanding) A∈Μ_n(ℤ) for which no digit set has ℤ[A, script D] = ℤ². However we show that for all expanding integer matrices A in dimensions 2 and 3, there exists some digit set script 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.