Slowly Divergent Trajectories for Weighted Singular Vectors in R^2

Abstract

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 $0<\nu<1$ and $\mu>0$, a separate power schedule gives Hausdorff dimension $s_w$ for the weighted uniform approximation set with rate $Q^{-1}\exp\bigl(-\mu(\log Q)^\nu\bigr)$, 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.