## Abstract

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 ⊂ R^{d}, d ≥ 2, with dim_{H}(E) > t_{k,d}, then the (_{2}^{k+1})-dimensional Lebesgue measure of T_{k}(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.

Original language | English |
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Pages (from-to) | 799-810 |

Number of pages | 12 |

Journal | Revista Matematica Iberoamericana |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - 2015 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© European Mathematical Society.

## Keywords

- Erdos-Falconer problems
- Mattila integral