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Let   $X$   be a quasi-projective variety over a number field, admitting (after passage to   $\mathbb {C}$  ) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of   $X$   are  sparse : the number of such points of height   $\leq B$   grows slower than any positive power of   $B$  . For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions.

Sparsity of Integral Points on Moduli Spaces of Varieties