Rigidity, Graphs and Hausdorff Dimension

A set of \(k+1\) points in Euclidean space is called a \((k+1)\)-point configuration. Two configurations are congruent if they equal up to an affine isometry. Given compact subset E \(\mathbb R^d\), \(d\ge 2\) Hausdorff dimension greater than \(d-\frac{1}{k+1}\) we prove that the Lebesgue measure noncongruent positive, for \(k>d\), complementing results [11] \(k\le d\).