四次Salem伯努利卷积的严格Hausdorff维数下降

中文摘要 AI 辅助

对于每一个四次Salem数β，我们证明等权伯努利卷积ν_{β⁻¹}的Hausdorff维数严格小于1。核心步骤是将几何分离频率处的大傅里叶系数转化为有限卷积的香农熵亏空；正三角乘积用于熵估计，代数范数控制每个空间单元中不同原子的数量。对于区间(1,2)内的两个四次Salem数，合适返回时刻的倒数差产生所需的傅里叶系数，再通过有限块覆盖得到维数界。该证明不依赖数值计算，单独的精确计算给出dim_Hν_{β₁⁻¹}<1-1.4·10⁻²³和dim_Hν_{β₂⁻¹}<1-7.8·10⁻²²，其中β₁<β₂是区间(1,2)内的两个参数，配套验证程序使用有理区间算术和Python标准库。

英文摘要

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 $\dimHν_{β_1^{-1}}<1-1.4\cdot10^{-23}$ and $\dimHν_{β_2^{-1}}<1-7.8\cdot10^{-22}$, 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.

发表机构

• Dalian University of Technology（大连理工大学）
• National Yang Ming Chiao Tung University（国立阳明交通大学）

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