  Monsky's theorem from 1970 states that a square cannot be dissected into an
odd number of triangles of the same area, but it does not give a lower bound
for the area differences that must occur.
  We extend Monsky's theorem to "constrained framed maps"; based on this we can
apply a gap theorem from semi-algebraic geometry to a polynomial area
difference measure and thus get a lower bound for the area differences that
decreases doubly-exponentially with the number of triangles. On the other hand,
we obtain the first superpolynomial upper bounds for this problem, derived from
an explicit construction that uses the Thue-Morse sequence.
