Full Dimensional Sets Without Given Patterns

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.