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 the density properties of rational points on the Kummer variety Km(Ar) 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 so…