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arXiv · 1306.3598

A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral

A. Greenleaf·A. Iosevich·B. Liu·E. Palsson
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Abstract

We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d) s_0(d)$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \cite{GI12} and in higher dimensions \cite{GGIP12}. In this paper, we improve upon those exponents, using a group-theoretic method that sheds new light on the classical approach to these problems. The key to our approach is a group action perspective which leads to natural and effective formulae related to the classical Mattila integral.

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BibTeXRIS

A. Greenleaf, A. Iosevich, B. Liu, E. Palsson. 2014-04-10. A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral. https://arxiv.org/abs/1306.3598

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