When does a Bernoulli convolution admit a spectrum

By Xin-Rong Dai. 2012

228 citations recorded by the source corpus.

Abstract In this paper, we solve a long-standing problem on Bernoulli convolutions. In particular, we show that the Bernoulli convolution μ ρ with contraction rate ρ ∈ ( 0 , 1 ) admits a spectrum if and only if ρ is the reciprocal of an even integer.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9d660c3055608873b87ce225eed70826ce8a9c75

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Xin-Rong Dai
2. Xinggang He
3. Chun-Kit Lai
4. Li-Xiang An
5. Xiaoye Fu
6. Qiyu Sun
7. Ka-Sing Lau
8. J. Miao

In Computer Vision and Pattern Recognition

1. Xinggang He
2. Xin-Rong Dai
3. Li-Xiang An
4. Chun-Kit Lai
5. Pablo Shmerkin
6. Qiyu Sun
7. Ka-Sing Lau
8. Xiaoye Fu

In Mathematical Physics

1. Xin-Rong Dai
2. Xinggang He
3. Chun-Kit Lai
4. Li-Xiang An
5. Xiaoye Fu
6. Qiyu Sun
7. Ka-Sing Lau
8. J. Miao

Citations in the corpus

• Spectral measures with arbitrary Hausdorff dimensions (2014, background): “Since then, various singular spectral measures on self-similar/self-affine fractal sets have been found, see for instance [3–6,9,11,15–17,19,21,22,24,26,27,30,34].”
• Spectrality of Self-Similar Tiles (2013, background): “itions (e.g., convexity). The problem also led to the study of the spectral property of the discrete integer tiles ([LW4], [PW]) and the Cantor-type singularly continuous measures ([JP], [ŁW], [HuL], [D], [DHS], [DHL]). In this paper, our main purpose is to initiate a study on the spectral property ”
• On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem (2019, background): “te convolution, Fourier frames, and sum of sine functions. The research of Lixiang An and Xiaoye Fu is supported by NSFC grant 11601175, 11401205. 1 2 LIXIANG AN, XIAOYE FU, AND CHUN-KIT LAI measures [9, 10, 11], as well as the convergence properties of the associated Fourier series [38, 39, 13] and”
• On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions (2013, background): “oof is a minor variant of the proof of Theorem 1.3 (using [7, Theorem 1.4]) and is left to the reader. As an example, we get: Corollary 1.5. If D is any finite set with at least two elements, and BD ⊂ [0,1] are the numbers whose continued fraction expansion has digits only in D, then Aλ + BD has posi”
• Weak convergence and spectrality of infinite convolutions (2022, background): “For the spectrality of infinite Bernoulli convolutions, Dai [11] showed that νλ is a spectral measure if and only if λ = 1/(2q) for some positive integer q, and then Dai, He, and Lau [12] generalized it to the case of N -Bernoulli convolution.”
• Spectral measures generated by arbitrary and random convolutions (2015, background): “[Dai12] Xin-Rong Dai.”
• Spectra of Cantor measures (2014, background): “In fact, a Bernoulli convolutionμ2,b with contraction ratio 1 /b is a spectral measure if and only if the contraction ratio is the reciproca l of an even integer [2].”