Existence of Many Large Orbits

A plaid polygon is called N-fat if it is not contained in any strip of width N. As a related notion, a plaid polygon is called N-long if it has diameter at least N. This chapter will prove Theorem 0.8. Section 22.2 studies equidistribution properties of the plaid PET map Φ‎A, as a function of A. Section 22.3 uses these equidistribution properties to show that the N-fat polygons essentially appear everywhere in the planar plaid model. The result is called the Ubiquity Lemma. Section 22.4 examines how the plaid model interacts with the grid of all lines of capacity at most K. Section 22.5 uses the Rectangle Lemma on many scales to show the existence of many distinct N-fat polygons. Section 2.6 discusses some properties of continued fractions and circle rotations. Finally, Section 22.7 proves the Grid Supply Lemma.