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

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We provide a near-complete classification of the Lorentz spaces Λφ for which the sequence {Sₙ}ₙ∈ ₙ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function φ, we identify Λφ:= L L L as the largest Lorentz space on which the lacunary Carleson operator is bounded as a map to L¹,∞. 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 L¹, being responsible for the unboundedness of the weak-L¹ norm of a "grand maximal counting function" associated with the mass levels.