Haar Bases forL2(imagen) and Algebraic Number Theory

We correct an error in the proof of Theorem 1.5 in Lagarias and Wang (J. Number Theory57, 1996, 181–197). We also give a strengthened necessary condition for the existence of a Haar basis of the specified kind for every integer matrixAthat has a given irreducible characteristic polynomialf(x) with f(0)=2. A. Potiopa (Masterʹs thesis, Siedlce University, 1997) found that the expanding polynomialg(x)=x4+x2+2 violates this necessary condition. Thus there exists a 4×4 expanding integral matrixAof determinant 2 and characteristic polynomialg(x) which has no Haar-type wavelet basis using an integer digit set imagesubset of or equal toimage4.