  A circle, centered at the origin and with radius chosen so that it has
non-empty intersection with the integer lattice $\mathbb{Z}^{2}$, gives rise to
a probability measure on the unit circle in a natural way. Such measures, and
their weak limits, are said to be attainable from lattice points on circles.
  We investigate the set of attainable measures and show that it contains all
extreme points, in the sense of convex geometry, of the set of all probability
measures that are invariant under some natural symmetries. Further, the set of
attainable measures is closed under convolution, yet there exist symmetric
probability measures that are not attainable. To show this, we study the
geometry of projections onto a finite number of Fourier coefficients and find
that the set of attainable measures has many singularities with a "fractal"
structure. This complicated structure in some sense arises from prime powers -
singularities do not occur for circles of radius $\sqrt{n}$ if $n$ is square
free.
