Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions
Abstract
For every quartic Salem number β, we prove that the equal-weight Bernoulli convolution νβ−1 has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in (1,2), reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give dimHνβ1−1<1−1.4·10−23 and dimHνβ2−1<1−7.8·10−22, where β1<β2 are the two parameters in (1,2). The accompanying verification programs use rational interval arithmetic and the Python standard library.
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BibTeX
@misc{cheng2026strict,
title = {{Strict Hausdorff-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.2610.12072},
eprint = {2610.12072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12072},
}
RIS
TY - UNPB
TI - Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions
AU - Guozheng Cheng
AU - Xiang Fang
AU - Xueqing Ma
AU - Hongli Zhang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12072
UR - https://arxiv.org/abs/2610.12072
AB - For every quartic Salem number β, we prove that the equal-weight Bernoulli convolution ν_β^−1 has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in (1,2), reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give dimHν_β_1^−1<1−1.4·10^−23 and dimHν_β_2^−1<1−7.8·10^−22, where β_1<β_2 are the two parameters in (1,2). The accompanying verification programs use rational interval arithmetic and the Python standard library.
ER -