On Kemnitz' Conjecture Concerning Lattice Points in the Plane

TL;DR

This paper proves Kemnitz' conjecture, extending a classical number theory result about subsets of integers to lattice points in the plane, demonstrating that certain configurations always exist.

Contribution

The paper provides the first proof of Kemnitz' conjecture, establishing a new combinatorial property of lattice points in the plane.

Findings

Proof of Kemnitz' conjecture confirmed

Extension of Erdős-Ginzburg-Ziv theorem to lattice points

New combinatorial result in additive number theory

Abstract

In 1961, P. Erd\H{o}s, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of integers, i.e., planar lattice points, usually referred to as Kemnitz' conjecture.