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摘要
We obtain nontrivial exponents for Erdos-Falconer type point configuration problems. Let T-k (E) denote the set of distinct congruent k-dimensional simplices determined by (k + 1)-tuples of points from E. For 1 <= k <= d, we prove that there exists a t(k),(d) < d such that, if E subset of R-d, d >= 2, with dim(H)(E) > t(k,d), then the ((k+1)(2))-dimensional Lebesgue measure of Tk (E) is positive. Results of this type were previously obtained for triangles in the plane (k = d = 2) in [8] and for higher k and d in [7]. We improve upon those exponents, using a group action perspective, which also sheds light on the classical approach to the Falconer distance problem.
- 出版日期2015
