Title

Two Problems on Cantor Set Arithmetic

Author(s)

Lauren Chen

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].

Location

Furman Hall 121

Start Date

3-28-2020 12:30 PM

Presentation Format

Oral and Written

Group Project

No

COinS
 
Mar 28th, 12:30 PM

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].