We prove several theorems concerning the exceptional sets of the Hilbert transform on the real line. In particular, it is proved that any null set is an exceptional set for the Hilbert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.
"On Exceptional Sets of the Hilbert Transform." Real Anal. Exchange 42 (2) 311 - 328, 2017. https://doi.org/10.14321/realanalexch.42.2.0311