How large dimension guarantees a given angle?

TL;DR

This paper investigates how the Hausdorff dimension of sets in Euclidean space is constrained by the absence of specific angles, revealing unique behaviors for certain angles like 0°, 90°, 120°, and 180°.

Contribution

It characterizes the maximal Hausdorff dimension of sets avoiding particular angles, highlighting special roles of certain angles in geometric measure theory.

Findings

Different angles impose different dimension constraints.

Angles 0°, 90°, 120°, and 180° are geometrically special.

The dimension bounds vary notably for these special angles.

Abstract

We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.