Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles

Monsky’s theorem from 1970 states that a square cannot be dissected into an odd number n 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 n. 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.