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We prove that the weak- $L^p$norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim <\#comment/>(p-<\#comment/>1{)}^{-<\#comment/>1}$as $p\to <\#comment/>1^{+}$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak- $L^p$type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space ${\mathrm{Q}\mathrm{A}}_{\mathrm{\infty <\#comment/>}}\left(w\right)$, which contains the weighted Antonov space $L\log <\#comment/>L\log <\#comment/>\log <\#comment/>\log <\#comment/>L(\mathbb{T};w)$, converge almost everywhere whenever $w\in <\#comment/>A_1$. This is an extension of the results of Antonov [Proceedings of the XXWorkshop on Function Theory (Moscow, 1995), 1996, pp. 187–196] and Arias De Reyna, where $w$must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near- $L^1$Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding relies on a newly developed smooth multi-frequency decomposition which, near the endpoint $p=1$, outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele [Math. Res. Lett. 17 (2010), pp. 529–545]. As a further example of application, we obtain a quantified version of the family of sparse bounds for the bilinear Hilbert transforms due to Culiuc, Ou and the first author.