  Let r > 0 be an integer. We present a sufficient condition for an abelian
variety A over a number field k to have infinitely many quadratic twists of
rank at least r, in terms of density properties of rational points on the
Kummer variety Km(A^r) of the r-fold product of A with itself. One consequence
of our results is the following. Fix an abelian variety A over k, and suppose
that for some r > 0 the Brauer-Manin obstruction to weak approximation on the
Kummer variety Km(A^r) is the only one. Then A has a quadratic twist of rank at
least r. Hence if the Brauer-Manin obstruction is the only one to weak
approximation on all Kummer varieties, then ranks of twists of any
positive-dimensional abelian variety are unbounded. This relates two
significant open questions.
