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Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket

  • Zheng-Yi Lu , Xin-Han Dong and Peng-Fei Zhang ORCID logo EMAIL logo
Published/Copyright: July 19, 2019

Abstract

Let μM,D be a self-affine measure generated by an expanding diagonal matrix MM3() with entries ρ1,ρ2,ρ3 and the digit set D={(0,0,0)t,(1,0,0)t,(0,1,0)t,(0,0,1)t}. In this paper, we prove that for any ρ1,ρ2,ρ3(1,), if ρ1,ρ2,ρ3{±x1r:x+,r+}, then L2(μM,D) contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of ρ1,ρ2,ρ3 that are in the set {±(pq)1r:p2+,q2+-1 and r+}. In particular, if ρ1,ρ2,ρ3{pq:p,q2+1}, then there exist at most 4 mutually orthogonal exponential functions in L2(μM,D), and the number 4 is the best possible.

MSC 2010: 42C05; 28A80

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11831007

Award Identifier / Grant number: 11571099

Funding statement: The research is supported in part by the NNSF of China (No.11831007, No.11571099). The first author is also supported by the Hunan Provincial Innovation Foundation for Postgraduate.

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Received: 2019-03-10
Published Online: 2019-07-19
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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