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We prove that the lacunary Carleson operator is bounded from $L \log L$ to$L^{1}$. This result is sharp.  The proof is based on two newly introduced concepts: 1) the\emph{time-frequency regularization of a measurable set} and 2) the\emph{set-resolution of the time-frequency plane at $0-$frequency}.  These two concepts will play the central role in providing a special tiledecomposition adapted to the interaction between the \emph{structure} of thelacunary Carleson operator and the corresponding \emph{structure} of a fixmeasurable set.  Another key insight of our paper is that it provides for the first time asimultaneous treatment of families of tiles with \emph{distinct} massparameters. This should be regarded as a fundamental feature/difficulty of theproblem of the pointwise convergence of Fourier Series near $L^1$, context inwhich, unlike the standard $L^p,\:p>1$ case, \emph{no decay} in the massparameter is possible.

The pointwise convergence of Fourier Series (II). Strong $L^1$ case for the lacunary Carleson operator