Dimensions of triangle sets

TL;DR

This paper establishes new lower bounds for the dimensions of triangle sets derived from compact sets in the plane, revealing how the original set's dimension influences the complexity of its triangle configurations.

Contribution

It provides novel dimension inequalities for triangle sets in \\mathbb{R}^2, including bounds for both Assouad and box dimensions under mild separation conditions.

Findings

\\dim_{A} \\Delta(F) \\geq \\frac{3}{2} \\dim_{A} F

If \\dim_{A} F > 1, then \\dim_{A} \\Delta(F) \\geq 1 + \\dim_{A} F

Improved bounds for \\dim_{A} \\Delta(F) when \\dim_{A} F > 4/3

Abstract

In this paper, we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^2$. In particular, we prove that for any compact set $F$ in $\mathbb{R}^2$, the triangle set $\Delta(F)$ satisfies \[ \dim_{\mathrm{A}} \Delta(F)\geq \frac{3}{2}\dim_{\mathrm{A}} F. \] If $\dim_{\mathrm{A}} F>1$ then we have \[ \dim_{\mathrm{A}} \Delta(F)\geq 1+\dim_{\mathrm{A}} F. \] If $\dim_{\mathrm{A}} F>4/3$ then we have the following better bound, \[ \dim_{\mathrm{A}} \Delta(F)\geq \min\left\{\frac{5}{2}\dim_{\mathrm{A}} F-1,3\right\}. \] Moreover, if $F$ satisfies a mild separation condition then the above result holds also for the box dimensions, namely, \[ \underline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\underline{\dim_{\mathrm{B}}} \Delta(F) \text{ and }\overline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\overline{\dim_{\mathrm{B}}} \Delta(F). \]