An uncountable subring of $\mathbb R$ with Hausdorff dimension zero

TL;DR

This paper constructs a subring of the real numbers that has Lebesgue measure zero and Hausdorff dimension zero, demonstrating a novel example of such a mathematical structure.

Contribution

It provides the first explicit example of an uncountable subring of  with Hausdorff dimension zero, expanding understanding of measure and dimension in algebraic structures.

Findings

Constructed an uncountable subring of  with measure zero.

Proved the subring has Hausdorff dimension zero.

Showed the subring's measure and dimension properties are distinct from typical algebraic sets.

Abstract

We construct a subring as mentioned in the title (hence this subring has Lebesgue measure zero).