The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin

We provide a near-complete classification of the Lorentz spaces ${\mathrm{\Lambda }}_{\phi }$ for which the sequence $\{S_n{\}}_{n\in \mathbb{N}}$ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function $\phi$, we identify ${\mathrm{\Lambda }}_{\phi }:=L\log \log L\log \log \log \log L$ as the \emph{largest} Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{1,\mathrm{\infty }}$. In particular, we disprove a conjecture stated by Konyagin in his 2006 ICM address. Our proof relies on a newly introduced concept of a “Cantor Multi-tower Embedding,” a special geometric configuration of tiles that can arise within the time-frequency tile decomposition of the Carleson operator. This geometric structure plays an important role in the behavior of Fourier series near $L^1$, being responsible for the unboundedness of the weak-$L^1$ norm of a “grand maximal counting function” associated with the mass levels.