Pointwise convergence of Fourier series (I). On a conjecture of Konyagin

We provide a near-complete classification of the Lorentz spaces Lambda(phi) for which the sequence {S-n}(n epsilon N) of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function phi, we identify Lambda(phi) := L log log L log log log log L as the largest Lorentz space on which the lacunary Carleson operator is bounded as a map to L-1,L-infinity. As a consequence, . we disprove a conjecture stated by Konyagin in his 2006 ICM address; . we provide a negative answer to an open question related to the Halo conjecture. 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.