Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation

Consider the action of \(a_t=\textrm{diag}(e^{nt},e^{-r_1(t)},\ldots ,e^{-r_n(t)})\in \textrm{SL}(n+1,{\mathbb {R}})\) $a_t=\text{diag}(e^{\mathit{nt}},e^{-r_1(t)},\dots ,e^{-r_n(t)})\in \text{SL}(n+1,\mathbb{R})$ , where \(r_i(t)\rightarrow \infty \) $r_i(t)\to \infty$ for each i, on the space of unimodular lattices in \({\mathbb {R}}^{n+1}\) ${\mathbb{R}}^{n+1}$ . We show that \(a_t\) $a_t$ -translates of segments of size \(e^{-t}\) $e^{-t}$ about all except countably many points of a nondegenerate smooth horospherical curve get equidistributed in the space as \(t\rightarrow \infty \) $t\to \infty$ . This result implies that the weighted Dirichlet approximation theorem cannot be improved for almost all points on any nondegenerate \(C^{2n}\) $C^{2n}$ curve in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ . These results extend the corresponding results for translates of fixed pieces of analytic curves due to Shah (2010) as well as those for uniform translates of shrinking curves due to Shah and Yang (2023), and answer some questions inspired by the work of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).