Non-spectral problem for a class of planar self-affine measures

Abstract

Listen

Translate article

The self-affine measure μ M , D corresponding to an expanding matrix M ∈ M n ( R ) and a finite subset D ⊂ R n is supported on the attractor (or invariant set) of the iterated function system { ϕ d ( x ) = M −1 ( x + d ) } d ∈ D . The spectral and non-spectral problems on μ M , D , including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μ M , D is to estimate the number of orthogonal exponentials in L 2 ( μ M , D ) and to find them. In the present paper we show that if a , b , c ∈ Z , | a | > 1 , | c | > 1 and a c ∈ Z ∖ ( 3 Z ) , M = [ a b 0 c ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) } , then there exist at most 3 mutually orthogonal exponentials in L 2 ( μ M , D ) , and the number 3 is the best. This extends several known conclusions. The proof of such result depends on the characterization of the zero set of the Fourier transform μ ˆ M , D , and provides a way of dealing with the non-spectral problem.