Spectrality And Non-spectrality Of A Class Of Self-affine Measures With Product Digit Sets

The self-affine measure ?M,D associated with an expansive matrix M ?Mn(Z)and a finite set D(?)Zn is uniquely determined by the self-affine identity with e-qual weight.In 1998,P.E.T.Jorgensen and S.Pedersen found the first example of self-affine measures which is a spectral measure,to promote the concept of spectral set.Since then,the research on the theory of spectral measure in recent years has become an important topic.Many mathematians have investigated the problem of spectrality and non-spectrality of self-affine measures.Based on their investigation,in this paper,the problem of the maximal number of mutually orthogonal exponen-tial functions,the spectrality and the non-spectrality for a class of planar self-affine measures are studied.The main results are as follows:(1)Consider the expansive integer matrix given by and the digit set is the maxhnal number of mutually orthogonal exponentials in L2(?M.D),is studied.It has been proven:there exist at most 12 mutually orthogonal exponentials in(2)By use of residue class of 3 and the feature of zero-point sets of the Fourier's transformation ?M,D,it is proven that for any expansive matrix with form and the above digit set D,if det(M)= ac—bd ? 3Z,then there exist an infinite family of orthogonal exponentials E(?)in L2(?M,D)with(?)(?)Zn.(3)For the expansive integer matrix M ? M2(Z)of(2)and the digit set D of(1),if a,b,c,d ? 3Z,then ?M,D is a spectral measure.