  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 tile
decomposition adapted to the interaction between the \emph{structure} of the
lacunary Carleson operator and the corresponding \emph{structure} of a fix
measurable set.
  Another key insight of our paper is that it provides for the first time a
simultaneous treatment of families of tiles with \emph{distinct} mass
parameters. This should be regarded as a fundamental feature/difficulty of the
problem of the pointwise convergence of Fourier Series near $L^1$, context in
which, unlike the standard $L^p,\:p>1$ case, \emph{no decay} in the mass
parameter is possible.
