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Tomas Serate Roos: Thick points for truncated random Fourier series

Time: Fri 2022-06-03 13.00 - 14.30

Location: KTH, Room 3418

Respondent: Tomas Serate Roos

Supervisor: Maurice Duits

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In this thesis we aim to give insight into the Hausdorff dimension of the thick points of a certain type of truncated Fourier series. The study of thick points has applications in areas such as conformal field theory, random matrices and even graph theory. This type of thick point problem has been studied before, but typically with the assumption that the random variables \(a_j\) have a standard complex normal distribution. We will drop this assumption here, taking \(a_j\) to be arbitrary rotationally invariant complex random variables, with mean 0, variance 1 and a bound on the moments. We present a proof for the upper bound of Hausdorff dimension of the sets of \(\alpha\)-thick points, finding that the result for the non-Gaussian case coincides with the known results for when \(a_j\) are taken to be Gaussian, and also give some ideas on how to approach the lower bound.