Abstract
We present a new method for establishing the “gap” property for finitely generated subgroups of
\mathrm{SU}(2)
, providing an elementary solution of Ruziewicz problem on
S^2
as well as giving many new examples of finitely generated subgroups of
\mathrm{SU}(2)
with an explicit gap. The distribution of the eigenvalues of the elements of the group ring
\mathbf{R}[\mathrm{SU}(2)]
in the
N
-th irreducible representation of
\mathrm{SU}(2)
is also studied. Numerical experiments indicate that for a generic (in measure) element of
\mathbf{R}[\mathrm{SU}(2)]
, the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for
N
even) and the GSE spacing law (for
N
odd) as
N\to \infty
; we establish several results in this direction. For certain special “arithmetic” (or
Ramanujan
) elements of
\mathbf{R}[\mathrm{SU}(2)]
the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.