### Full Dimensional Sets Without Given Patterns

Péter Maga
Source: Real Anal. Exchange Volume 36, Number 1 (2010), 79-90.

#### Abstract

We construct a $d$ Hausdorff dimensional compact set in $\R^d$ that does not contain the vertices of any parallelogram. We also prove that for any given triangle ($3$ given points in the plane) there exists a compact set in $\R^2$ of Hausdorff dimension $2$ that does not contain any similar copy of the triangle. On the other hand, we show that the set of the $3$-point patterns of a $1$-dimensional compact set of $\R$ is dense.

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Primary Subjects: 28A78
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1300108086
Zentralblatt MATH identifier: 06032475
Mathematical Reviews number (MathSciNet): MR3016405

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