On the measure of products from the middle-third Cantor set

TL;DR

This paper establishes bounds on the Lebesgue measure of the product set of the middle-third Cantor set, using a novel subdivision approach that improves convergence of approximation formulas.

Contribution

It introduces a new subdivision method for the Cantor set that yields faster convergence in measuring the product set’s Lebesgue measure.

Findings

Derived upper and lower bounds for the measure of the product set.

Developed a subdivision technique that accelerates approximation convergence.

Enhanced understanding of measure properties of Cantor set products.

Abstract

We prove upper and lower bounds for the Lebesgue measure of the set of products $xy$ with $x$ and $y$ in the middle-third Cantor set. Our method is inspired by Athreya, Reznick and Tyson, but a different subdivision of the Cantor set provides a more rapidly converging approximation formula.