Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation (Q6624134)

This paper further develops the theory of equidistribution for translates of curves, segments of curves, and shriking segments of curves in various settings of homogeneous dynamics building in particular on earlier work by the authors, see [\textit{N. A. Shah}, J. Am. Math. Soc. 23, No. 2, 563--589 (2010; Zbl 1200.11055); \textit{N. A. Shah} and \textit{P. Yang}, J. Mod. Dyn. 19, 947--965 (2023; Zbl 1533.37010)]. \N\NPart of the development here is to allow for non-uniformity in the translation in each coordinate in the following sense. For the action of \(a_t=\operatorname{diag}(\exp(nt),\exp(-r_1(t)),\dots,\exp(-r_n(t))) \in \mathrm{SL}_{n+1}(\mathbb{R})\) with \(r_i(t)\to\infty\) as \(t\to\infty\) on the space of unimodular lattices in \(\mathbb{R}^{n+1}\) it is shown that translates under \(a_t\) of segments of size \(\exp(-t)\) about all but countably many points of a non-degenerate smooth horospherical curve equidistribute as \(t\to\infty\). As with earlier results of this type, one of the applications is to show non-improvability of certain Diophantine approximation theorems. \N\NThe (very) broad outline of the argument is to first verify a non-divergence criterion as used by \textit{S. G. Dani} and \textit{G. A. Margulis} [Adv. Sov. Math. 16, 91--137 (1993; Zbl 0814.22003)] (and many others subsequently) and show that the measures do not escape to infinity, then show that any limiting measure must be invariant under a non-trivial unipotent subgroup, allowing Ratner's theorem to be applied to show that the limiting measure has enough algebraic structure to enable linearization techniques to be used to give a more precise description of the limiting measures.