Lattice-tiling properties of integral self-affine functions

✍ Scribed by M.N. Kolountzakis

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 184 KB
• Volume: 10
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(97)00073-6

✦ Synopsis

Let A be a d x d expanding integer matrix and p : Z d ---* C be absolutely summable and satisfy ~~.~ez~ p(x) = t det A I. A function f e L 1 (R d) is called an integral self-a•ne function for the pair (A,p) if it satisfies the functional equation f(A-lx) = ~~.z/ez a p(y)f(x -y), a.e. (x).

We prove that for such a function there is always a sublattice A of Z d such that f tiles R a with A with weight w = ]Z d : A] -lfR df. That is 5-].;~hf(x-A) = w, a.e. (x). The lattice A _C Z d is the smallest A-invariant sublattice of Z d that contains the support of p. This generalizes results of Lagarias and Wang [1] and others, which were obtained for f and p which are indicator functions of compact sets. The proofs use Fourier Analysis.