# Two Problems on Cantor Set Arithmetic

## School Name

Dutch Fork High School

## Grade Level

12th Grade

## Presentation Topic

Mathematics

## Presentation Type

Mentored

## Abstract

We ﬁnd the cardinalities of the solution sets to the polynomial equations c = a + b and c = a − b on variants of the Cantor set. We also compute examples for the equation c = ab. A previous theorem states f(C × C) = [0, 1] for the Cantor set C where f(x, y) = x2y. Our second problem generalizes this to f = xαy for α in the range log 3/2log 2 ≤ α ≤ 2. We also explore the case when α is greater than 2. We consider the expansion of f(Cn × Cn) for a few small n, where Cn is the nth iteration of the Cantor set, to ﬁnd intervals of α > 2 such that f(C × C) does not cover the entire interval [0, 1].

## Recommended Citation

Chen, Lauren, "Two Problems on Cantor Set Arithmetic" (2020). *South Carolina Junior Academy of Science*. 289.

https://scholarexchange.furman.edu/scjas/2020/all/289

## Location

Furman Hall 121

## Start Date

3-28-2020 12:30 PM

## Presentation Format

Oral and Written

## Group Project

No

Two Problems on Cantor Set Arithmetic

Furman Hall 121

We ﬁnd the cardinalities of the solution sets to the polynomial equations c = a + b and c = a − b on variants of the Cantor set. We also compute examples for the equation c = ab. A previous theorem states f(C × C) = [0, 1] for the Cantor set C where f(x, y) = x2y. Our second problem generalizes this to f = xαy for α in the range log 3/2log 2 ≤ α ≤ 2. We also explore the case when α is greater than 2. We consider the expansion of f(Cn × Cn) for a few small n, where Cn is the nth iteration of the Cantor set, to ﬁnd intervals of α > 2 such that f(C × C) does not cover the entire interval [0, 1].