On the Set of Distances Between the Points of a Carathéodory Linearly Measurable Plane Point Set

Besicovitch, A. S.; Miller, Daniel

doi:10.1112/plms/s2-50.4.305

Cited by 8 publications

(8 citation statements)

References 3 publications

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“…Namely, if S(K) is not dense in S 1 , then K is contained in a rectifiable curve by [28,Lemma 15.13]. Then as H 1 (K) > 0, a result of Besicovitch and Miller [4] gives that D(K) contains an interval, and in particular dim D(K) = 1. Let E be the lower semicontinuous function given by Q, defined above in Proposition 2.7, and let ε > 0.…”

Section: Proofs Of the Distance Set Resultsmentioning

confidence: 99%

Scaling scenery of (×m,×n) invariant measures

Ferguson

Fraser²,

Sahlsten³

2015

Advances in Mathematics

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Abstract. We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x, y) → (mx mod 1, ny mod 1) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman-Shmerkin and Ferguson-Jordan-Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford-McMullen, Lalley-Gatzouras and Barański class and all planar self-similar sets.

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Section: Proofs Of the Distance Set Resultsmentioning

confidence: 99%

Scaling scenery of (×m,×n) invariant measures

Ferguson

Fraser²,

Sahlsten³

2015

Advances in Mathematics

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“…Let s ≥ 1, and let K ⊂ B(0, 1) be an s-AD-regular set with H s (K) > 0. If s = 1 and K contains a non-trivial rectifiable part, then a result of Besicovitch and Miller [1] from 1948 tells us immediately that D(K) has positive length: in particular dim p D(K) = dim B D(K) = 1. So, one may assume that either s > 1, or K is purely 1-unrectifiable.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

On the distance sets of Ahlfors–David regular sets

Orponen

2017

Advances in Mathematics

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I prove that if ∅ = K ⊂ R 2 is a compact s-Ahlfors-David regular set with s ≥ 1, then dimp D(K) = 1, where D(K) := {|x − y| : x, y ∈ K} is the distance set of K, and dimp stands for packing dimension.The same proof strategy applies to other problems of similar nature. For instance, one can show that if ∅ = K ⊂ R 2 is a compact s-Ahlfors-David regular set with s ≥ 1, then there exists a point x0 ∈ K such that dimp K · (K − x0) = 1. Specialising to product sets, one derives the following sum-product corollary: if A ⊂ R is a non-empty compact s-Ahlfors-David regular set with s ≥ 1/2, thenfor some a1, a2 ∈ A. In particular, dimp[AA + AA − AA − AA] = 1. In all of the results mentioned above, compactness can be relaxed to boundedness and H s -measurability, if packing dimension is replaced by upper box dimension.

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“…It remains to prove Proposition 2. This is a generalization of the smoothing process used by Besicovitch and Miller [1]. A sketch of the proof is given; further details can be found in [7].…”

Section: £ D(b) ^ C{a(k)-5} > A(g\x) I = Lmentioning

confidence: 99%

“…If/is injective when C is a simple curve, and if A(C) < oo then C is rectifiable.A. S. Besicovitch and D. S. Miller[1] have shown that all simple rectifiable curves in the plane have the Steinhaus property for distance sets; and A. S. Besicovitch and S. J. Taylor [2] have shown, by giving a counterexample, that not all simple rectifiable curves in a general metric space have this property.…”

mentioning

confidence: 99%

On Extensions of the Steinhaus Theorem for Distance Sets and Difference Sets

Boardman

1972

Journal of the London Mathematical Society

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On the Set of Distances Between the Points of a Carathéodory Linearly Measurable Plane Point Set

Cited by 8 publications

References 3 publications

Scaling scenery of (×m,×n) invariant measures

Scaling scenery of (×m,×n) invariant measures

On the distance sets of Ahlfors–David regular sets

On Extensions of the Steinhaus Theorem for Distance Sets and Difference Sets

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