  In the present paper we establish bounds for the size of the spectral gap for
actions of algebraic groups on certain homogeneous spaces. Our approach is
based on estimating operator norms of suitable averaging operators, and we
develop techniques for establishing both upper and lower bounds for such norms.
We shall show that this analytic problem is closely related to the arithmetic
problem of establishing bounds on the discrepancy of distribution for rational
points on algebraic group varieties. As an application, we show how to
establish an effective bound for property $\tau$ of congruence subgroups of
arithmetic lattices in algebraic groups which are forms of $SL(2)$, using
estimates in intrinsic Diophantine approximation which follow from
Heath-Brown's analysis of rational points on 3-dimensional quadratic surfaces.
