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In this paper, we study the distribution of integral points on parametricfamilies of affine homogeneous varieties. By the work of Borel andHarish-Chandra, the set of integral points on each such variety consists offinitely many orbits of arithmetic groups, and we establish an asymptoticformula (on average) for the number of the orbits indexed by their Siegelweights. In particular, we deduce asymptotic formulas for the number ofinequivalent integral representations by decomposable forms and by norm formsin division algebras, and for the weighted number of equivalence classes ofintegral points on sections of quadratic surfaces. Our arguments use theexponential mixing property of diagonal flows on homogeneous spaces.

Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows