数字集的谱性与谱自仿测度

Spectrality of digit sets and spectral self-affine measures

本文主要确定与谱自仿测度μ_(M,D)相关的数字集D的谱性.研究所涉及的谱性问题与Dutkay、Han和Jorgensen的一个猜测密切相关.此猜测表明,在一维情形下,如果μ_(M,D)是谱自仿测度,则相应的数字集D总是一个谱集.对于一个自仿测度μ_(M,D),本文获得使数字集D具有谱性的一些条件,为这个猜测的成立提供了依据.另外,本文所得的结果在某些情形下也推广许多已知的相应结果.

The present paper determines the spectrality of digit set D relating to a spectral self-affine measure PM, D. This is motivated by a conjecture of Dutkay, Han and Jorgensen. The conjecture states that D is always a spectral set if μM,D is a spectral measure in the dimension n = 1. For a self-affine measure μM,D, we obtain several conditions for the digit set D to be a spectral set. The result here provides some supportive evidence for the conjecture. It also generalizes the corresponding known result in a certain case. Keywords self-afflne measure, spectrality, compatible pair, digit set