On distance sets, box-counting and Ahlfors regular sets
We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance se...
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2017 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/77253 |
| Acesso em linha: | http://hdl.handle.net/11336/77253 |
| Access Level: | acceso abierto |
| Palavra-chave: | Distance sets Box dimension Ahlfors-regular sets CP-processes https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance sets of A have lower box-counting dimension 1. We also show that if A,B ⊂ R 2 have Hausdorff dimension greater than 1 and A is Ahlfors-regular, then the set of distances between A and B has modified lower box-counting dimension 1, which taking B = A improves Orponen’s result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections. |
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