Rational Polygons: Odd Compression Ratio and Odd Plane Coverings

TL;DR

This paper investigates properties of rational polygons, demonstrating that the area of points covered an odd number of times by translates is bounded away from zero, and introduces an odd plane cover construction.

Contribution

It establishes a lower bound on the area of points with odd coverage and constructs an odd cover of the plane using translates of rational polygons.

Findings

The area of points with odd coverage is bounded away from zero.

An odd cover of the plane by translates of P is constructed.

The results depend only on the shape of P.

Abstract

Let P be a polygon with rational vertices in the plane. We show that for any finite odd-sized collection of translates of P, the area of the set of points lying in an odd number of these translates is bounded away from 0 by a constant depending on P alone. The key ingredient of the proof is a construction of an odd cover of the plane by translates of P. That is, we establish a family F of translates of P covering (almost) every point in the plane a uniformly bounded odd number of times.