MathDB

Equilateral triangle with vertices near lattices points

Source: 2017 Korean Winter Program Practice Test 1 Day 1 #4

January 18, 2017

number theorygeometry

Problem Statement

For a point

P

on the plane, denote by

\lVert P \rVert

the distance to its nearest lattice point. Prove that there exists a real number

L > 0

satisfying the following condition:
For every

\ell > L

, there exists an equilateral triangle

ABC

with side-length

\ell

and

\lVert A \rVert, \lVert B \rVert, \lVert C \rVert < 10^{-2017}