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Ciprian Demeter, Michael Lacey, Christoph Thiele and I have just uploaded our joint paper, “The Walsh model for $M_2^*$ Carleson” to the arXiv. This paper (which was recently accepted for publication in Revista Iberoamericana) establishes a simplified model for the key estimate (the “$M_2^*$ Carleson estimate”) in another (much longer) paper of ours on the return times theorem of Bourgain, in which the Fourier transform is replaced by its dyadic analogue, the Walsh-Fourier transform. This model estimate is established by the now-standard techniques of time-frequency analysis: one decomposes the expression to be estimated into a sum over tiles, and then uses combinatorial stopping time arguments into group the tiles into trees, and the trees into forests. One then uses (phase-space localised, and frequency-modulated) versions of classical Calderòn-Zygmund theory (or in this particular case, a certain maximal Fourier inequality of Bourgain) to control individual trees and forests, and sums up over the trees and forests using orthogonality methods (excluding an exceptional set if necessary).

Rather than discuss time-frequency analysis in detail here, I thought I would dwell instead on the return times theorem, and sketch how it is connected to the $M_2^*$ Carleson estimate; this is a more complicated version of the “$M_2$ Carleson estimate”, which is an estimate which is logically equivalent to Carleson’s famous theorem (and its extension by Hunt) on the almost everywhere convergence of Fourier series.