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Non-spectrality of Moran measures with four digits. (English) Zbl 1499.28017

The authors study spectral measures \(\mu\), compactly supported Borel probability measures on \(\mathbb{R}^n\) for which there exists a countable set \(\Lambda\) satisfying \[E_{\Lambda}:=\{e^{2\pi i\langle \lambda,x\rangle}: \lambda\in\Lambda\} \text{ is an orthonormal basis for }L^2(\mu).\] In this case \(\Lambda\) is called a spectrum of \(\mu\) and \((\mu,\Lambda)\) is called a spectral pair.
The attention is focused on Moran measures \(\mu_{\{M_k\},\{D_k\}}\) with compact support, generated by a Moran iterated function system, cf. [J. E. Hutchinson, Indiana Univ. Math. J. 30, 713–747 (1981; Zbl 0598.28011)]: a family of contraction functions \(\{\phi_{k,D_k}(x)=M_k^{-1}(x+d): d\in D_k\}_{k=1}^{\infty}\), where \(\{M_k\}_{k=1}^{\infty}\subset M_n(\mathbb{R})\) is a sequence of expanding matrices (all eigenvalues have modulus strictly greater than one and \(\{D_k\}_{k=1}^{\infty}\) is a sequence of finite digit sets of \(\mathbb{R}^n\)).
Assuming that \(T=\sum_{k=1}^{\infty}\,M_1^{-1}M_2^{-1}\cdots M_k^{-1}D_k\) is a compact subset of \(\mathbb{R}^n\), the sequence of measures \[\mu_k=\delta_{M_1^{-1}D_1}\ast\delta_{M_1^{-1}M_2^{-1}D_2}\ast\cdots\ast\delta_{M_1^{-1}M_2^{-1}\cdots M_k^{-1}D_k}\] tends to a Borel probability measure with compact support \(T\), denoted by \(\mu_{\{M_k\},\{D_k\}}\).
The main results, given in the Introduction, are:
Theorem 1.3: Let the limit Moran measure be given as above. Then \(L_2(\mu_{\{M_k\},\{D_k\}})\) contains an infinite orthogonal set of exponential functions if and only if \(\mathrm{det}(M_k)\in 2\mathbb{Z}\) for infinitely many \(k\).
Theorem 1.4: Let the limit Moran measure be given as above. If \(\mathrm{det}(M_k)\in 2\mathbb{Z}+1\) for any \(k\geq 1\), then there exist at most \(4\) mutually orthogonal functions in \(L_2(\mu_{\{M_k\},\{D_k\}})\), and the number \(4\) is the best.

MSC:

28A80 Fractals
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Citations:

Zbl 0598.28011
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Full Text: DOI

References:

[1] An, L. X.; He, X. G., A class of spectral Moran measures, J. Funct. Anal., 266, 343-354 (2014) · Zbl 1303.28009
[2] An, L. X.; He, X. G.; Lau, K. S., Spectrality of a class of infinite convolutions, Adv. Math., 283, 362-376 (2015) · Zbl 1323.28007
[3] An, L. X.; Fu, X. Y.; Lai, C. K., On spectral Cantor-Moran measures and a variant of Bourgain’s sum of sine problem, Adv. Math., 349, 84-124 (2019) · Zbl 1416.42005
[4] Chen, M. L.; Liu, J. C., The cardinality of orthogonal exponentials of planar selfaffine measures with three-element digit sets, J. Funct. Anal., 277, 135-156 (2019) · Zbl 1414.28016
[5] Chen, M. L.; Liu, J. C.; Su, J.; Wang, X. Y., Spectrality of a class of Moran measures, Canad. Math. Bull., 63, 366-381 (2020) · Zbl 1437.28010
[6] Dai, X. R., When does a Bernoulli convolution admit a spectrum?, Adv. Math., 231, 1681-1693 (2012) · Zbl 1266.42012
[7] Dai, X. R.; He, X. G.; Lai, C. K., Spectral property of Cantor measures with consecutive digits, Adv. Math., 242, 187-208 (2013) · Zbl 1277.28009
[8] Dai, X. R.; He, X. G.; Lau, K. S., On spectral N-Bernoulli measures, Adv. Math., 259, 511-531 (2014) · Zbl 1303.28011
[9] Deng, Q. R., On the spectra of Sierpinski-type self-affine measures, J. Funct. Anal., 270, 4426-4442 (2016) · Zbl 1337.42028
[10] Deng, Q. R.; Wang, X. Y., On the spectra of self-affine measures with three digits, Anal. Math., 45, 267-289 (2019) · Zbl 1438.42048
[11] Dutkay, D. E.; Jorgensen, P. E T., Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z., 256, 801-823 (2007) · Zbl 1129.28006
[12] Dutkay, D. E.; Haussermann, J.; Lai, C. K., Hadamard triples generate self-affine spectral measures, Trans. Amer. Math. Soc., 371, 1439-1481 (2019) · Zbl 1440.42030
[13] Dutkay, D. E.; Lai, C. K., Spectral measures generated by arbitrary and random convolutions, J. Math. Pures Appl., 107, 183-204 (2017) · Zbl 1364.42011
[14] Fu, Y. S.; He, X. G.; Wen, Z. X., Spectra of Bernoulli convolutions and random convolutions, J. Math. Pures Appl., 116, 105-131 (2018) · Zbl 1400.42005
[15] Fuglede, B., Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal., 16, 101-121 (1974) · Zbl 0279.47014
[16] He, L.; He, X. G., On the Fourier orthonormal bases of Cantor-Moran measures, J. Funct. Anal., 272, 1980-2004 (2017) · Zbl 1357.28010
[17] Hu, T. Y.; Lau, K. S., Spectral property of the Bernoulli convolutions, Adv. Math., 219, 554-567 (2008) · Zbl 1268.42044
[18] Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J., 30, 713-747 (1981) · Zbl 0598.28011
[19] Jorgensen, P. E T.; Pedersen, S., Dense analytic subspaces in fractal L^2-spaces, J. Anal. Math., 75, 185-228 (1998) · Zbl 0959.28008
[20] Li, J. L., Non-spectrality of self-affine measures on the spatial Sierpinski gasket, J. Math. Anal. Appl., 432, 1005-1017 (2015) · Zbl 1321.28018
[21] Li, J. L., Non-spectral problem for a class of planar self-affine measures, J. Funct. Anal., 255, 3125-3148 (2008) · Zbl 1211.28007
[22] Li, J. L., Analysis of μ_M,D-orthogonal exponentials for the planar four-element digit sets, Math. Nachr., 287, 297-312 (2014) · Zbl 1291.28009
[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) · Zbl 1366.28009
[24] Liu, J. C.; Zhang, Y.; Wang, Z. Y.; Chen, M. L., Spectrality of generalized Sierpinski-type self-affine measures, Appl. Comput. Harmon. Anal., 55, 129-148 (2021) · Zbl 1472.28007
[25] Kolountzakis, M.; Matolcsi, M., Complex Hadamard matrices and the spectral set conjecture, Collect. Math., 57, 281-291 (2006) · Zbl 1134.42313
[26] Kolountzakis, M.; Matolcsi, M., Tiles with no spectra, Forum Math., 18, 519-528 (2006) · Zbl 1130.42039
[27] Ramsey, F. P., On a problem of formal logic, Proc. London Math. Soc., 30, 264-286 (1930) · JFM 55.0032.04
[28] Strichartz, R., Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math., 81, 209-238 (2000) · Zbl 0976.42020
[29] Su, J.; Liu, Y.; Liu, J. C., Non-spectrality of the planar self-affine measures with four-element digit sets, Fractals, 27, 1950115 (2019) · Zbl 1434.28034
[30] Tao, T., Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett., 11, 251-258 (2004) · Zbl 1092.42014
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