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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 $\lesssim (p-1)^{-1}$as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where thesame 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 atthe endpoint $p=1$. In particular, we obtain that the Fourier series offunctions from the weighted Arias de Reyna space $ QA_{\infty}(w) $, whichcontains the weighted Antonov space $L\log L\log\log\log L(\mathbb T; w)$,converge almost everywhere whenever $w\in A_1$. This is an extension of theresults of Antonov 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 smoothmulti-frequency decomposition which, near the endpoint $p=1$, outperforms theabstract Hilbert space approach of past works, including the seminal one byNazarov, Oberlin and Thiele. As a further example of application, we obtain aquantified version of the family of sparse bounds for the bilinear Hilberttransforms due to Culiuc, Ou and the first author.

The weak-type Carleson theorem via wave packet estimates