The Point-to-Set Principle and the Dimensions of Hamel Bases

TL;DR

This paper demonstrates that every real number in [0,1] can be realized as the Hausdorff dimension of a Hamel basis of the reals over rationals, using a novel proof involving algorithmic fractal dimension and the point-to-set principle.

Contribution

It introduces a new proof technique combining computability theory and fractal geometry to analyze the dimensions of algebraic structures.

Findings

Every real in [0,1] is the Hausdorff dimension of some Hamel basis.

The proof employs algorithmic fractal dimension and the point-to-set principle.

The approach bridges computability theory and geometric measure theory.

Abstract

We prove that every real number in [0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals over the field of rationals. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension--a computability-theoretic construct--and the point-to-set principle of J. Lutz and N. Lutz (2018).