## Real Analysis Exchange

### Full Dimensional Sets Without Given Patterns

Péter Maga

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 14 March 2011

https://projecteuclid.org/euclid.rae/1300108086

Mathematical Reviews number (MathSciNet)
MR3016405

Zentralblatt MATH identifier
1246.28005

Subjects
Primary: 28A78: Hausdorff and packing measures

#### Citation

Maga, Péter. Full Dimensional Sets Without Given Patterns. Real Anal. Exchange 36 (2010), no. 1, 79--90. https://projecteuclid.org/euclid.rae/1300108086

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