A combinatorial proof of Marstrand's Theorem for products of regular Cantor sets

TL;DR

This paper provides a combinatorial proof of Marstrand's Theorem specifically for products of regular Cantor sets with certain smoothness, demonstrating that their projections have positive measure when the sum of their dimensions exceeds one.

Contribution

It introduces a novel combinatorial proof for Marstrand's Theorem applicable to products of regular Cantor sets with C^{1+a} smoothness.

Findings

Proves that projections of product Cantor sets have positive measure under specified conditions.

Establishes a combinatorial approach as an alternative to classical analytical proofs.

Extends understanding of geometric measure theory for fractal sets.

Abstract

In 1954 Marstrand proved that if K is a subset of R^2 with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^{1+a}, a>0, for which the sum of their Hausdorff dimension is greater than 1.