The Quasi-Isomorphism Theorem

This chapter proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.

The Projections of Convex Lattice Sets of Points in E2

Can one determine a centrally symmetric lattice polygon by its projections? In 2005, Gardner et al. proposed the above discrete version of Aleksandrov’s projection theorem. In this paper, we define a coordinate matrix for a centrally symmetric convex lattice set and suggest an algorithm to study this problem.