  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 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 $ QA_{\infty}(w) $, which
contains 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 the
results 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 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. 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.
