  The main result of this paper is the following. Given countably many
multivariate polynomials with rational coefficients and maximum degree $d$, we
construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does
not contain finite point configurations corresponding to the zero sets of the
given polynomials.
  Given a set $E\subset \R^n$, we study the angles determined by three points
of $E$. The main result implies the existence of a compact set in $\R^n$ of
Hausdorff dimension $n/2$ which does not contain the angle $\pi/2$. (This is
known to be sharp if $n$ is even.) We show that there is a compact set of
Hausdorff dimension $n/8$ which does not contain an angle in any given
countable set. We also construct a compact set $E\subset \R^n$ of Hausdorff
dimension $n/6$ for which the set of angles determined by $E$ is Lebesgue null.
  In the other direction, we present a result that every set of sufficiently
large dimension contains an angle $\epsilon$ close to any given angle.
