  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.
