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 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.Comment: 32 pages, 22 figures. Version v1: Sections 3.1-3.3 have been restructured; a new Section 8 on even dissections has been added. Version v2 includes a correction in Section 7.5 that was not completely carried out in the journal versio