Non-spectral problem for the planar self-affine measures with decomposable digit sets

Abstract

In this paper, we consider the non-spectral problem for the planar self-affine measures M,D generated by an expanding integer matrix M ∈ M 2 (ℤ) and a finite digit set3 (mod ℤ 2 ) and gcd(det(M), 3) = 1 , then there exist at most max{17, 9 + 8} mutually orthogonal exponential functions in L 2 ( M,D ) , where = max{r ∶ 3 r |( 1 4 − 2 3 )}. KeywordsSelf-affine measure • Spectral measure • Orthogonal exponential functions • Fourier transform Mathematics Subject Classification 42C05 • 46C05 Tusi Mathematical Research GroupCommunicated by Baruch Solel.