Falconer distance problem, additive energy and Cartesian products

Iosevich, Alex; Liu, Bochen

doi:10.5186/aasfm.2016.4135

Cited by 10 publications

(6 citation statements)

References 9 publications

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“…As remarked in [22], 4/3 appears to be the limit of these methods. Nevertheless, Iosevich and Liu [11] recently obtained an improvement for a large class of cartesian products.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On distance sets, box-counting and Ahlfors-regular sets

Shmerkin¹

2017

Discreteanal.

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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 ⊂ 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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“…As remarked in [22], 4/3 appears to be the limit of these methods. Nevertheless, Iosevich and Liu [11] recently obtained an improvement for a large class of cartesian products.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On distance sets, box-counting and Ahlfors-regular sets

Shmerkin¹

2017

Discreteanal.

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“…Observe that when dim A = dim B, the assumption dim A + dim B/2 > 2 in Theorem 4.2 becomes dim A > 4/3 and is the same as Wolff's. In [15] Iosevich and Liu improved distance set results of the time for product sets with rather simple arguments. For the most recent, and so far the best-known, distance set results, see [12], [4], and [5].…”

Section: Distance Sets and Measuresmentioning

confidence: 99%

Hausdorff dimension and projections related to intersections

Mattila¹

2022

Publicacions Matemàtiques

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“…Various partial results on distance sets have recently been proved, among others, by Iosevich and Liu [IL1], [IL2], Lucá and Rogers [LR], Orponen [O5] and Shmerkin [S3], [S4].…”

Section: General Intersectionsmentioning

confidence: 99%