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DIMENSIONS OF TRIANGLE SETS

Part of: Classical measure theory Discrete geometry

Published online by Cambridge University Press:  18 December 2018

Han Yu
Han Yu*
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, U.K. email [email protected]
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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 $\unicode[STIX]{x1D6E5}(F)$ satisfies

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant {\textstyle \frac{3}{2}}\dim _{\text{A}}F.\end{eqnarray}$$
If $\dim _{\text{A}}F>1$, then we have
$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant 1+\dim _{\text{A}}F.\end{eqnarray}$$
 If $\dim _{\text{A}}F>4/3$, then we have the following better bound:
$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant \min \{{\textstyle \frac{5}{2}}\dim _{\text{A}}F-1,3\}.\end{eqnarray}$$
 Moreover, if $F$ satisfies a mild separation condition, then the above result holds also for the box dimensions, namely,
$$\begin{eqnarray}\text{}\underline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\text{}\underline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F)\quad \text{and}\quad \overline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\overline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F).\end{eqnarray}$$

MSC classification

Primary: 28A80: Fractals
Secondary: 52C10: Erdős problems and related topics of discrete geometry 52C15: Packing and covering in $2$ dimensions 52C35: Arrangements of points, flats, hyperplanes
Type
Research Article
Information
Mathematika , Volume 65 , Issue 2 , 2019 , pp. 311 - 332
Copyright
Copyright © University College London 2018 

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