A manifestation of the Uncertainty Principle through Fourier coefficients

The uncertainty principle in Harmonic analysis captures the fact that spatial localization and Fourier decay cannot occur simultaneously beyond certain limits, and many deep results in Harmonic analysis reflect this fundamental interplay in various ways. In this talk, I will present a summable analogue of a classical theorem dating back to the work of O. Ivashev-Musatov in 1958, later refined by T. W. Körner in 2003. More specifically, we shall identify thresholds connecting two phenomena: localization measured in terms of meagre support, and weak regularity measured by weighted average Fourier decay.