Splittable Coverings of The Integers

School Name

Governor's School for Science & Mathematics

Grade Level

12th Grade

Presentation Topic

Mathematics

Presentation Type

Mentored

Mentor

Mentor: Gregory Clark, University of South Carolina

Abstract

Let a tile be defined as a non-empty subset of the integers. The concept of decomposable coverings as discussed by J. Pach and G. Toth can be extended to these integer tiles. In this context, a decomposable covering is defined as any covering of Z that can be partitioned into two distinct coverings of Z. Furthermore, we show that any finite integer tile can be used to construct a decomposable covering and provide bounds for the densities of such coverings. These results are expanded upon in the case of three-element integer tiles.

Location

Wall 205

Start Date

3-25-2017 11:15 AM

Presentation Format

Oral and Written

Group Project

No

COinS
 
Mar 25th, 11:15 AM

Splittable Coverings of The Integers

Wall 205

Let a tile be defined as a non-empty subset of the integers. The concept of decomposable coverings as discussed by J. Pach and G. Toth can be extended to these integer tiles. In this context, a decomposable covering is defined as any covering of Z that can be partitioned into two distinct coverings of Z. Furthermore, we show that any finite integer tile can be used to construct a decomposable covering and provide bounds for the densities of such coverings. These results are expanded upon in the case of three-element integer tiles.