Strict correlation-dimension drop for quartic Salem Bernoulli convolutions
Abstract
For every quartic Salem number β∈(1,2), we prove that the Bernoulli convolution νβ−1 with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an L2 density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give 0.99999<D2(νβ1−1)<1−10−65, 0.99999<D2(νβ2−1)<1−10−75, where β1<β2 are the two quartic Salem numbers in (1,2).
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BibTeX
@misc{cheng2026strict,
title = {{Strict correlation-dimension drop for quartic Salem Bernoulli convolutions}},
author = {Guozheng Cheng and Xiang Fang and Xueqing Ma and Hongli Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09902},
eprint = {2610.09902},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09902},
}
RIS
TY - UNPB
TI - Strict correlation-dimension drop for quartic Salem Bernoulli convolutions
AU - Guozheng Cheng
AU - Xiang Fang
AU - Xueqing Ma
AU - Hongli Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09902
UR - https://arxiv.org/abs/2610.09902
AB - For every quartic Salem number β∈(1,2), we prove that the Bernoulli convolution ν_β^−1 with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an L^2 density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give 0.99999<D_2(ν_β_1^−1)<1−10^−65, 0.99999<D_2(ν_β_2^−1)<1−10^−75, where β_1<β_2 are the two quartic Salem numbers in (1,2).
ER -