\(R^2\)中加权奇异向量的缓慢发散轨迹
Slowly Divergent Trajectories for Weighted Singular Vectors in R^2
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中文总结 AI 辅助
研究\(R^2\)中加权奇异向量，证明满足特定条件的\(w\) - 奇异向量集豪斯多夫维数为\(s_w = 2 - (1 + w_1)^{-1}\)。对不同参数给出加权均匀逼近集维数，用自仿射构造等方法直接证明下包络定理。
中文摘要 AI 辅助
设\(w=(w_1,w_2)\)满足\(w_1>w_2>0\)且\(w_1 + w_2 = 1\)。对于每个\(f:[0,\infty)\to(0,\infty)\)且\(f(t)\to0\)的函数，证明了加权最短向量函数满足\(W_x(t)\geq\log f(t)\)（对所有足够大的\(t\)）的\(w\) - 奇异向量集的豪斯多夫维数\(s_w = 2 - (1 + w_1)^{-1}\)，等于\(\text{Sing}_w(2)\)的全豪斯多夫维数。对于每个\(0 < \nu < 1\)和\(\mu > 0\)，单独的幂次方案给出了加权均匀逼近集的豪斯多夫维数\(s_w\)，其速率为\(Q^{-1}\exp(-\mu(\log Q)^\nu)\)，而下包络定理给出了其在\(\text{Sing}_w(2)\)中的补集的相同维数。通过将廖 - 石 - 索兰 - 塔马姆的自仿射构造适应于可变的返回时间序列，并使用空分母窗口论证来控制中间尖点偏移，而无需进一步修剪树，给出了下包络定理的直接证明。
英文摘要
Let $w=(w_1,w_2)$ satisfy $w_1>w_2>0$ and $w_1+w_2=1$. For every function $f:[0,\infty)\to(0,\infty)$ with $f(t)\to0$, we prove that the set of $w$-singular vectors whose weighted shortest-vector function satisfies $W_x(t)\ge\log f(t)$ for all sufficiently large $t$ has Hausdorff dimension $s_w=2-(1+w_1)^{-1}$, equal to the full Hausdorff dimension of $\operatorname{Sing}_w(2)$. For every $0<ν<1$ and $μ>0$, a separate power schedule gives Hausdorff dimension $s_w$ for the weighted uniform approximation set with rate $Q^{-1}\exp\bigl(-μ(\log Q)^ν\bigr)$, whereas the lower-envelope theorem gives the same dimension for its complement in $\operatorname{Sing}_w(2)$. We give a direct proof of the lower-envelope theorem by adapting the self-affine construction of Liao--Shi--Solan--Tamam to a variable sequence of return times and using an empty-denominator-window argument to control intermediate cusp excursions without further pruning the tree.
