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].
