Skip to main content
Log in

Non-spectral problem for the planar self-affine measures with decomposable digit sets

  • Original Paper
  • Published:
Annals of Functional Analysis Aims and scope Submit manuscript

Abstract

In this paper, we consider the non-spectral problem for the planar self-affine measures \(\mu _{M,D}\) generated by an expanding integer matrix \(M\in M_2({\mathbb {Z}})\) and a finite digit set

$$\begin{aligned} D=\{(0,0)^t, (\alpha _1,\alpha _2)^t, (\alpha _3, \alpha _4)^t \} \oplus k(\alpha _1\alpha _4-\alpha _2\alpha _3){\mathfrak {D}}, \end{aligned}$$

where \(k\in {\mathbb {Z}}\backslash \{0\}, \alpha _i\in {\mathbb {Z}}\;(1\le i\le 4)\) and \({\mathfrak {D}\subset \mathbb{R}^2}\) is a finite integer digit set. Let \(Z(m_D)=\{x\in {\mathbb {R}}^2: \sum _{d\in D} e^{2\pi i \langle d, x\rangle }=0\}\) and \(\mathring{E_3^2}:=\frac{1}{3}\{(l_1, l_2)^t: l_1, l_2\in \mathbb{N},0\le l_1, l_2\le 2\}\backslash \{0\}\). We prove that if \(\alpha _1\alpha _4-\alpha _2\alpha _3\ne 0\), \(Z(m_{{\mathfrak {D}}})\subset \mathring{E_{3 }^2} \pmod {{\mathbb {Z}}^2}\) and \(\gcd (\det (M), 3)=1\), then there exist at most \(\max \{17,9^{\eta }+8\}\) mutually orthogonal exponential functions in \(L^2(\mu _{M,D})\), where \(\eta =\max \{r: 3^r| (\alpha _1\alpha _4-\alpha _2\alpha _3)\}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. An, L.X., He, X.G., Tao, L.: Spectrality of the planar Sierpinski family. J. Math. Anal. Appl. 432, 725–732 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bellissard, J., Bessis, D., Moussa, P.: Chaotic states of almost periodic Schrödinger operators. Phys. Rev. Lett. 49, 701–704 (1982)

    Article  MathSciNet  Google Scholar 

  3. Chen, M.L., Liu, J.C.: The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets. J. Funct. Anal. 277, 135–156 (2019)

    Article  MathSciNet  Google Scholar 

  4. Dai, X.R.: When does a Bernoulli convolution admit a spectrum? Adv. Math. 231, 1681–1693 (2012)

    Article  MathSciNet  Google Scholar 

  5. Dai, X.R., He, X.G., Lai, C.K.: Spectral structure of Cantor measures with consecutive digits. Adv. Math. 242, 187–208 (2013)

    Article  MathSciNet  Google Scholar 

  6. Dai, X.R., He, X.G., Lau, K.S.: On spectral \(N\)-Bernoulli measures. Adv. Math. 259, 511–531 (2014)

    Article  MathSciNet  Google Scholar 

  7. Deng, Q.R., Lau, K.S.: Sierpinski-type spectral self-similar measures. J. Funct. Anal. 269, 1310–1326 (2015)

    Article  MathSciNet  Google Scholar 

  8. Dutkay, D., Haussermann, J.: Number theory problems from the harmonic analysis of a fractal. J. Number Theory 159, 7–26 (2016)

    Article  MathSciNet  Google Scholar 

  9. Dutkay, D.E., Haussermann, J., Lai, C.K.: Hadamard triples generate self-affine spectral measures. Trans. Am. Math. Soc. 371, 1439–1481 (2019)

    Article  MathSciNet  Google Scholar 

  10. Dutkay, D.E., Jorgensen, P.E.T.: Analysis of orthogonality and of orbits in affine iterated function systems. Math. Z. 256, 801–823 (2007)

    Article  MathSciNet  Google Scholar 

  11. Dutkay, D.E., Jorgensen, P.E.T.: Fourier frequencies in affine iterated function systems. J. Funct. Anal. 247(1), 110–137 (2007)

    Article  MathSciNet  Google Scholar 

  12. Emme, J.: Spectral measure at zero for self-similar tilings. Mosc. Math. J. 17, 35–49 (2017)

    Article  MathSciNet  Google Scholar 

  13. Feng, D.J., Wang, Y.: On the structures of generating iterated function systems of Cantor sets. Adv. Math. 222, 1964–1981 (2009)

    Article  MathSciNet  Google Scholar 

  14. Fu, Y.S., Wen, Z.X.: Spectral property of a class of Moran measures on \({\mathbb{R}}\). J. Math. Anal. Appl. 430, 572–584 (2015)

    Article  MathSciNet  Google Scholar 

  15. Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)

    Article  MathSciNet  Google Scholar 

  16. Hu, T.Y., Lau, K.S.: Spectral property of the Bernoulli convolutions. Adv. Math. 219, 554–567 (2008)

    Article  MathSciNet  Google Scholar 

  17. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)

    Article  MathSciNet  Google Scholar 

  18. Jorgensen, P.E.T., Pedersen, S.: Dense analytic subspaces in fractal \(L^2\)-spaces. J. Anal. Math. 75, 185–228 (1998)

    Article  MathSciNet  Google Scholar 

  19. Kolountzakis, M.N., Matolcsi, M.: Complex Hadamard matrices and the spectral set conjecture. Collect. Math. 57(Extra), 281–291 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Kolountzakis, M.N., Matolcsi, M.: Tiles with no spectra. Forum Math. 18, 519–528 (2006)

    Article  MathSciNet  Google Scholar 

  21. Li, J.L.: Non-spectral problem for a class of planar self-affine measures. J. Funct. Anal. 255, 3125–3148 (2008)

    Article  MathSciNet  Google Scholar 

  22. Li, J.L.: Spectrality of a class of self-affine measures with decomposable digit sets. Sci. China Math. 55, 1229–1242 (2012)

    Article  MathSciNet  Google Scholar 

  23. Liu, J.C., Dong, X.H., Li, J.L.: Non-spectral problem for the planar self-affine measures. J. Funct. Anal. 273, 705–720 (2017)

    Article  MathSciNet  Google Scholar 

  24. Tao, T.: Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11, 251–258 (2004)

    Article  MathSciNet  Google Scholar 

  25. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications I. Springer, New York (1985)

    Book  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous referees for their valuable suggestions and comments leading to the improvement of our manuscript. The research is supported in part by the NNSF of China (No. 11831007) and a project supported by Scientific Research Fund of Hunan Provincial Education Department (No. 19C0579).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhi-Min Wang.

Additional information

Communicated by Baruch Solel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, ZM. Non-spectral problem for the planar self-affine measures with decomposable digit sets. Ann. Funct. Anal. 11, 1287–1297 (2020). https://doi.org/10.1007/s43034-020-00086-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s43034-020-00086-6

Keywords

Mathematics Subject Classification

Navigation