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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 paperare: (1) A one-side extension of Frostman's theorem, which connects the rate ofdecay of Fourier transform of a distribution with the Hausdorff dimension ofthe support; (2) A construction of compacts of "critical" size, which supportdistributions (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.

Singular distributions, dimension of support, and symmetry of Fourier transform