## Abstract

We show that if a finite point set

• There is a

• There is a circle

This provides evidence for two conjectures of Erd˝os. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

*P*⊆ R^{2 }has the fewest congruence classes of triangles possible, up to a constant*M,*then at least one of the following holds.• There is a

*σ*> 0 and a line*l*which contains Ω(|*P*|*) points of*^{σ}*P*. Further, a positive proportion of*P*is covered by lines parallel to*l*each containing Ω(|*P*|*) points of*^{σ}*P*.• There is a circle

*γ*which contains a positive proportion of*P*.This provides evidence for two conjectures of Erd˝os. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

Original language | English |
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Pages (from-to) | 155–178 |

Number of pages | 24 |

Journal | Combinatorica |

Volume | 44 |

Issue number | 1 |

Early online date | 12 Oct 2023 |

DOIs | |

Publication status | Published - Feb 2024 |

### Bibliographical note

Publisher Copyright:© The Author(s) 2023.

## Keywords

- Erdos Distinct Distances
- Triangles
- additive energy