Jean-Philippe Labbé, Günter Rote, and Günter M. Ziegler:

Area difference bounds for dissections of a square into an odd number of triangles

Abstract

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.