  We study the "Fourier symmetry" of measures and distributions on the circle,
in relation with the size of their supports. The main results of this paper
are: (1) A one-side extension of Frostman's theorem, which connects the rate of
decay of Fourier transform of a distribution with the Hausdorff dimension of
the support; (2) A construction of compacts of "critical" size, which support
distributions (even pseudo-functions) with anti-analytic part belonging to l^2.
We also give examples of non-symmetry which may occur for measures with "small"
support. A number of open questions are stated.
