Year: 2025
Type: article
Abstract: We prove that, for every norm on \(\mathbb{R}^d\) and every \(E \subseteq \mathbb{R}^d\), the Hausdorff dimension of the distance set of \(E\) with respect to that norm is at least \(\dim_{\rm H} E - (d-1)\). An explicit construction follows, demonstrating that this bound is sharp for every polyhedral norm on \(\mathbb{R}^d\). The techniques of algorithmic complexity theory underlie both the computations and the construction.
Source: Real Analysis Exchange
Authors: Iqra Altaf, Ryan Bushling, Bobby Wilson
Institutions: University of Chicago, University of Washington
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4408854348