Characterization of universal Lebesgue-point convergence subsequences

Characterize the increasing sequences of positive integers for which the Cesàro means of subsequential Fourier partial sums converge at every Lebesgue point of every function in L1(T).

Background

The paper studies convergence of the Cesàro means T_m^a f(x) = m^{-1} sum_{k=1}^m S_{a_k}f(x) at Lebesgue points of integrable functions. Earlier work established sufficient or necessary conditions for particular classes of subsequences, including convex sequences, polynomial subsequences, and certain lacunary sequences, but these results do not provide a complete classification of all increasing sequences.

The paper disproves the sufficiency part of Belinsky’s conjecture by constructing a strictly convex increasing sequence satisfying a polynomial growth bound and an L1(T) function with a Lebesgue point at which the Cesàro means are unbounded. Consequently, the broader problem of identifying exactly which increasing sequences guarantee convergence at every Lebesgue point remains unresolved.