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The cardinality of μM,D‐orthogonal exponentials for the planar four digits

  • Jing-Cheng Liu , Yao Liu , Ming-Liang Chen and Sha Wu EMAIL logo
Published/Copyright: May 19, 2021

Abstract

In this work, we study the non-spectrality of the self-affine measure μM,D generated by an expanding integer matrix MM2() with det(M)2 and the integer digit set

D={(0,0)t,(α1,α2)t,(β1,β2)t,(-α1-β1,-α2-β2)t}

with α1β2-α2β10. Let η=max{s:2s|(α1β2-α2β1)}. We show that if 0η2, then L2(μM,D) contains at most 22(η+1) mutually orthogonal exponential functions, and the number 22(η+1) is the best. However, the number is strictly less than 22(η+1) if η3, and it is related to the order of the matrix M.

MSC 2010: 28A80; 42C05; 46C05

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12071125

Award Identifier / Grant number: 11831007

Award Identifier / Grant number: 2019JJ20012

Funding statement: The research is supported in part by the NNSF of China (nos. 12071125 and 11831007) and the Hunan Provincial NSF (no. 2019JJ20012).

Acknowledgements

The authors would like to thank the referee for his/her many valuable comments and suggestions.

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Received: 2021-01-18
Revised: 2021-04-11
Published Online: 2021-05-19
Published in Print: 2021-07-01

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