一类具有连续数字集测度的谱性

Spectrality of Some Measures withConsecutive Digits

摘要: 设μ_ρ,D_N,{n_k}是由以下离散测度的无穷卷积定义的Borel概率测度

Abstract: μ_ρ,D_N,{n_k}=δ_ρⁿ¹D_N*δ_ρⁿ²D_N*δ_ρⁿ³D_N*…，其中0<ρ<1,N ≥1且D_N = {0,1,2,…,N - 1}，{n_k}_k=1^∞是一个严格递增的正整数序列,且sup_k≥1{n_k+1-n_k}<∞. 本文证明了若μ_ρ,D_N,{n_k}是谱测度当且仅当存在一个自然数r使得N|ρ^-r且对所有的k ≥1，n_k+1-n_kρ,D_N,{n_k} be the Borel probability measure defined by the infinite convolution of the following discrete measures μ_ρ,D_N,{n_k}=δ_ρⁿ¹D_N*δ_ρⁿ²D_N*δ_ρⁿ³D_N*…，where0<ρ<1, N ≥1 andD_N = {0,1,2,…,N - 1}, {n_k}_k=1^∞ is a strictly increasing sequence of positive integers, and sup_k≥1{n_k+1-n_k}<∞. This paper demonstrates that μ_ρ,D_N,{n_k} is a spectral measure if and only if there exists a natural number r such that N|ρ^-r and for all k ≥1，n_k+1-n_k

文章引用：刘玉凡, 郑鹏辉. 一类具有连续数字集测度的谱性[J]. 理论数学, 2025, 15(11): 24-31. https://doi.org/10.12677/PM.2025.1511266

参考文献

[1]	Fuglede, B. (1974) Commuting Self-Adjoint Partial Differential Operators and a Group The- oretic Problem. Journal of Functional Analysis, 16, 101-121.[CrossRef]
[2]	Tao, T. (2004) Fuglede's Conjecture Is False in 5 and Higher Dimensions. Mathematical Re- search Letters, 11, 251-258. [Google Scholar] [CrossRef]
[3]	Farkas, B., Matolcsi, M. and Mora, P. (2006) On Fuglede's Conjecture and the Existence of Universal Spectra. Journal of Fourier Analysis and Applications, 12, 483-494.[CrossRef]
[4]	Kolountzakis, M.N. and Matolcsi, M. (2006) Tiles with No Spectra. Forum Mathematicum, 18, 519-528. [Google Scholar] [CrossRef]
[5]	Lev, N. and Matolcsi, M. (2022) The Fuglede Conjecture for Convex Domains Is True in All Dimensions. Acta Mathematica, 228, 385-420.[CrossRef]
[6]	He, X., Lai, C. and Lau, K. (2013) Exponential Spectra in L2(μ). Applied and Computational Harmonic Analysis, 34, 327-338. [Google Scholar] [CrossRef]
[7]	Dutkay, D.E. and Lai, C. (2014) Uniformity of Measures with Fourier Frames. Advances in Mathematics, 252, 684-707. [Google Scholar] [CrossRef]
[8]	Lai, C. (2011) On Fourier Frame of Absolutely Continuous Measures. Journal of Functional Analysis, 261, 2877-2889. [Google Scholar] [CrossRef]
[9]	Fan, A., Fan, S., Liao, L. and Shi, R. (2019) Fuglede's Conjecture Holds in Qp. Mathematische Annalen, 375, 315-341. [Google Scholar] [CrossRef]
[10]	Malikiosis, R.D. and Kolountzakis, M.N. (2018) Fuglede's Conjecture on Cyclic Groups of Order pnq. Discrete Analysis.[CrossRef]
[11]	Shi, R.-X. (2019) Fuglede's Conjecture Holds on Cyclic Groups Zpqr. Discrete Analysis, 14, 14.
[12]	Strichartz, R.S. (2000) Mock Fourier Series and Transforms Associated with Certain Cantor Measures. Journal d'Analyse Mathematique, 81, 209-238.[CrossRef]
[13]	Dai, X. (2012) When Does a Bernoulli Convolution Admit a Spectrum? Advances in Mathe- matics, 231, 1681-1693.[CrossRef]
[14]	Dai, X., He, X. and Lau, K. (2014) On Spectral N-Bernoulli Measures. Advances in Mathe- matics, 259, 511-531. [Google Scholar] [CrossRef]
[15]	An, L., He, X. and Lau, K. (2015) Spectrality of a Class of Infinite Convolutions. Advances in Mathematics, 283, 362-376. [Google Scholar] [CrossRef]
[16]	An, L., He, X. and Li, H. (2015) Spectrality of Infinite Bernoulli Convolutions. Journal of Functional Analysis, 269, 1571-1590. [Google Scholar] [CrossRef]
[17]	Wu, H.-H. and Zheng, P.-H. Spectrality of Some Infinite Convolutions on R. Submitted.
[18]	An, L.-X., Li, Q. and Zhang, M.-M. Characterization of Spectral Cantor-Moran Measures with Consecutive Digits. Preprint.
[19]	Jorgensen, P.E.T. and Pedersen, S. (1998) Dense Analytic Subspaces in Fractal L2-Spaces. Journal d'Analyse Mathematique, 75, 185-228. [Google Scholar] [CrossRef]
[20]	Dai, X., He, X. and Lau, K. (2014) On Spectral N-Bernoulli Measures. Advances in Mathe- matics, 259, 511-531. [Google Scholar] [CrossRef]
[21]	Wu, H.-H. and Liu, J.-C. (2022) Construction of a Class of Spectral Measures. Forum Mathe- maticum, 34, 1507-1517.