  Given two varieties V,W in the n-fold product of modular curves, we answer
affirmatively a question (formulated by Shou-Wu Zhang's AIM group) on whether
the set of points in V that are Hecke translations of some point on W is dense
in V. We need to make some (necessary) assumptions on the dimensions of V,W but
for instance, when V is a divisor and W is a curve, no further assumptions are
needed. We also examine the necessity of our assumptions in the case of
unlikely intersections and show that, contrary to exceptions, two curves in a
high dimensional space over a finite field can intersect infinitely often up to
Hecke translations.
