Abstract
A set of \(k+1\) points in Euclidean space is called a \((k+1)\)-point configuration. Two configurations are congruent if they are equal up to an affine isometry. Given a compact subset E of \(\mathbb R^d\), \(d\ge 2\) of Hausdorff dimension greater than \(d-\frac{1}{k+1}\) we prove that the Lebesgue measure of noncongruent \((k+1)\)-point configurations in E is positive, for \(k>d\), complementing the results of [11] for \(k\le d\).
The second and fourth listed authors were partially supported by the NSA Grant H98230-15-1-0319. The third listed author was partially supported by the Simons Foundation Collaboration Grant No. 422190.
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Chatzikonstantinou, N., Iosevich, A., Mkrtchyan, S., Pakianathan, J. (2021). Rigidity, Graphs and Hausdorff Dimension. In: Nathanson, M.B. (eds) Combinatorial and Additive Number Theory IV. CANT 2020. Springer Proceedings in Mathematics & Statistics, vol 347. Springer, Cham. https://doi.org/10.1007/978-3-030-67996-5_5
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