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The purpose of this article is twofold: to prove a pointwise equidistributiontheorem with an error rate for almost smooth functions, which strengthens themain result of Kleinbock, Shi and Weiss (2017); and to obtain aL\'evy-Khintchin theorem for weighted best approximations, which extends themain theorem of Cheung and Chevallier (2019).  To do so, we employ techniques from homogeneous dynamics and the methodsdeveloped in the work of Cheung-Chevallier (2019) and Shapira-Weiss (2022).

Pointwise equidistribution for almost smooth functions with an error rate and Weighted L\'evy-Khintchin theorem