#### Title

Splittable Coverings of The Integers

#### School Name

Governor's School for Science & Mathematics

#### Grade Level

12th Grade

#### Presentation Topic

Mathematics

#### Presentation Type

Mentored

#### 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.

#### Recommended Citation

Miyasaki, Sydney, "Splittable Coverings of The Integers" (2017). *South Carolina Junior Academy of Science*. 150.

http://scholarexchange.furman.edu/scjas/2017/all/150

#### Location

Wall 205

#### Start Date

3-25-2017 11:15 AM

#### Presentation Format

Oral and Written

#### Group Project

No

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.

## Mentor

Mentor: Gregory Clark, University of South Carolina