Approximate embedding of large polygons into $Z^2$

TL;DR

This paper demonstrates that large scaled polygons can be approximately embedded into the integer lattice $Z^2$ through a suitable rotation, with the approximation error arbitrarily small, extending prior results with a concise proof.

Contribution

The paper provides a short, self-contained proof that large scaled polygons can be rotated to approximate the integer lattice within any desired precision.

Findings

Existence of rotations aligning scaled polygons with $Z^2$ within $ ext{eps}$

Extension of Ziegler's 2006 result to broader settings

Simplified proof approach for approximate lattice embedding

Abstract

Let $Z^2$ denote the standard lattice in the plane $R^2$. We prove that given a finite subset $S\subset R^2$ and $\eps>0$, then for all sufficiently large dilations $t>0$ there exists a rotation $\rho\colon R^2\to R^2$ around the origin such that $\dist(\rho(tz),Z^2)<\eps$, for all $z\in S$. The result, in a larger generality, has been proved in 2006 by Tamar Ziegler (improving earlier results by Furstenberg, Katznelson, Weiss). The proof presented in the paper is short and self-contained.