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We show that under the action of$\mathrm{diag}(e^{nt},e^{-r_1(t)},\ldots,e^{-r_n(t)})\in\mathrm{SL}(n+1,\mathbb{R})$,where $r_i(t)\to\infty$, on the space of unimodular lattices in$\mathbb{R}^{n+1}$, the translates of any fixed-sized piece of a`non-degenerate' smooth curve, or a shrinking piece of size $e^{-t}$ aboutalmost any point of the curve, get equidistributed in the space as$t\to\infty$. From this, it follows that the weighted Dirichlet approximationtheorem cannot be improved for almost all points on any non-degenerate $C^{2n}$curve in $\mathbb{R}^n$. This result extends the corresponding result foranalytic curves due to Shah (2009) and answers some questions inspired by thework of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).

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