A different approach to the Fraenkel Conjecture for low $n$ values

TL;DR

This paper introduces a novel method for addressing Fraenkel's conjecture on partitioning integers into rational Beatty sequences with equal numerators, providing a new proof for the case when n=4.

Contribution

It offers a new approach to Fraenkel's conjecture and proves the n=4 case for equal numerators using this method.

Findings

New proof for the n=4 case of Fraenkel's conjecture

Introduces a novel approach to partitioning integers into Beatty sequences

Enhances understanding of the structure of rational Beatty sequences

Abstract

We present a new approach to deal with Fraenkel's conjecture, which describes how the integers can be partitioned into sets of rational Beatty sequences, in the case where the numerators of the moduli are equal. We use this approach to give a new proof of the known $n=4$ case when the numerators are equal.