四分Cantor测度的谱性质

Spectral properties of the one-fourth Cantor measure

自仿测度μM,D谱性质的研究始于四分Cantor测度μ4(即M=4,D={0,2}的情形).在长期从事谱集研究的基础上,Jorgensen和Pedersen在1998年首次发现μ4是一个具有谱性质的分形测度,其谱Λ(M,S)与和谐对(M^(-1)D,S)密切相关,其中S={0,1}.近年来的研究表明,对于某些奇数l,数乘集合lΛ(M,S)也是测度μ4的谱.这使得测度μ4的一些谱具有较强的稀疏性.本文重点对具有上述性质的奇数l进行讨论.利用数论中同余关系和有限群中元素的阶的性质,得到当l分别为素数、素数幂和素数乘积时,lΛ(M,S)为谱的判别依据,改进推广Dutkay等人的工作.

The research on the spectrality of self-affine measures μM,D starts with the one-fourth Cantor measure μ4（i.e., the case of M = 4, D = {0, 2}）. Based on the previous work for the spectral sets, Jorgensen and Pedersen first observed in 1998 that the fractal measure μ4 is spectral, moreover, its spectrum Λ（M, S） has close elations with the compatible pair（M-1D, S）, where S = {0, 1}. The recent studies on this subject illustrate that or certain odd integer l, the set lΛ（M, S） is also a spectrum for μ4. This is rather striking because some spectra or the measure μ4 are thinning. In the present paper, we mainly characterize the number l which has the above property. Three cases on the l, i.e., the cases when l is a prime, l is a power of the prime, and l is a product of primes, are considered respectively. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on l such that lΛ（M, S） is a spectrum for μ4. The result here extends he corresponding result of Dutkay and Jorgensen.