AI 中文总结
本文通过分析线性 Khintchine 不等式的依赖关系，发现维度超6时的相变现象并改进斜率常数，得到线性阈值函数一阶傅里叶权重的改进下界，推进了 O'Donnell 猜想。
AI 中文摘要
Khintchine 不等式为独立 Rademacher 随机变量加权和的期望绝对值提供了一个下界。在经典情形下，当权重向量的范数为1时，该下界为常数，仅在一类简单的极值构型中达到等号。De、Diakonikolas 和 Servedio（2013）提出的改进版本——称为线性 Khintchine 不等式——通过建立一个依赖于权重向量与极值集距离的线性下界，强化了上述结论。本文对该距离的依赖关系进行了精细分析。我们的结果揭示了改进速率的相变现象：当维度超过6时，随着权重向量偏离极小值点，下界会发生突变。此外，我们还改进了线性 Khintchine 不等式中的斜率常数。由此，我们得到了线性阈值函数（LTF）的一阶傅里叶权重的改进下界：$\boldsymbol{W}^{\leq 1}[\text{LTF}] \geq 0.53317$，这标志着向 O'Donnell 的猜想迈出了一步。
英文摘要
The Khintchine inequality provides a lower bound on the expected absolute value of a weighted sum of independent Rademacher random variables. In the classical setting, when the weight vector has unit norm, this lower bound is a constant, with equality attained only for a simple family of extremal configurations. A refined version due to De, Diakonikolas, and Servedio (2013) -- referred to as the \emph{linear Khintchine inequality} -- strengthens this by establishing a lower bound that depends linearly on the distance of the weight vector from the extremal set. In this paper, we present a refined analysis of this dependence on the weight vector. Our results reveal a phase transition in the rate of improvement: when the dimension exceeds six, the lower bound undergoes an abrupt change as the weight vector deviates from the minimizer. Additionally, we improve the slope constant in linear Khintchine inequality. As a consequence, we establish an improved lower bound on the degree-1 Fourier weight for linear threshold functions $\mathbf{W}^{\leq 1}[\mathrm{LTF}] \geq 0.53317$, marking progress towards a conjecture of O'Donnell.
