Correction theorems and the uncertainty principle

In 1977, F. G. Aroutyunyan proved that every continuous function on the unit circle, after a change on a set of arbitrarily small measure, acquires a uniformly convergent Fourier series and large gaps in the spectrum. Later, this version of the classical Men'shov correction theorem underwent several generalizations and refinements (partly, this was done by the author). In the talk we will review some of these results. The emphasis will be on the borders for the validity of the uncertainty principle (“a function and its Fourier transform cannot be too small simultaneously”) and on sharp estimates in correction theorems similar to those mentioned above.