How large dimension guarantees a given angle?

We study the following two problems:(1) Given $n\ge 2$ and $\al$, how largeHausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not containthree points that form an angle $\al$? (2) Given $\al$ and $\de$, how largeHausdorff dimension can a compact subset $A$ of a Euclidean space have if $A$does not contain three points that form an angle in the $\de$-neighborhood of$\al$? Some angles ($0,60^\circ,90^\circ, 120^\circ, 180^\circ$) turn out tobehave differently than other $\al\in[0,180^\circ]$.