Geodesic
On distance sets, box-counting and Ahlfors-regular sets
Discrete analysis (2017) Cet article a éte moissonné depuis la source Scholastica

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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 $A,B\subset\mathbb{R}^2$ have Hausdorff dimension $>1$ 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.
Publié le : 
@article{DAS_2017_a11,
     author = {Pablo Shmerkin},
     title = {On distance sets, box-counting and {Ahlfors-regular} sets},
     journal = {Discrete analysis},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2017_a11/}
}
TY  - JOUR
AU  - Pablo Shmerkin
TI  - On distance sets, box-counting and Ahlfors-regular sets
JO  - Discrete analysis
PY  - 2017
UR  - http://geodesic.mathdoc.fr/item/DAS_2017_a11/
LA  - en
ID  - DAS_2017_a11
ER  - 
%0 Journal Article
%A Pablo Shmerkin
%T On distance sets, box-counting and Ahlfors-regular sets
%J Discrete analysis
%D 2017
%U http://geodesic.mathdoc.fr/item/DAS_2017_a11/
%G en
%F DAS_2017_a11
Pablo Shmerkin. On distance sets, box-counting and Ahlfors-regular sets. Discrete analysis (2017). http://geodesic.mathdoc.fr/item/DAS_2017_a11/