Hausdorff Measure

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Li Feng - One of the best experts on this subject based on the ideXlab platform.

Zuoling Zhou - One of the best experts on this subject based on the ideXlab platform.

Baoguo Jia - One of the best experts on this subject based on the ideXlab platform.

Boris Solomyak - One of the best experts on this subject based on the ideXlab platform.

  • The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure
    Further Developments in Fractals and Related Fields, 2013
    Co-Authors: Yuval Peres, Boris Solomyak
    Abstract:

    In an earlier work, joint with R. Kenyon, we computed the Hausdorff dimension of the “multiplicative golden mean shift” defined as the set of all reals in [0, 1] whose binary expansion (x k ) satisfies x k x 2k = 0 for all k ≥ 1. Here we show that this set has infinite Hausdorff Measure in its dimension. A more precise result in terms of gauges in which the Hausdorff Measure is infinite is also obtained.

  • the sharp Hausdorff Measure condition for length of projections
    Proceedings of the American Mathematical Society, 2005
    Co-Authors: Yuval Peres, Boris Solomyak
    Abstract:

    In a recent paper, Pertti Mattila asked which gauge functions ' have the property that for any Borel set AR 2 with Hausdorff Measure H ' (A) > 0, the projection of A to almost every line has positive length. We show that finiteness of R 1 0 '(r) r2 dr, which is known to be sufficient for this property, is also necessary for regularly varying '. Our proof is based on a random construction adapted to the gauge function.

  • the sharp Hausdorff Measure condition for length of projections
    arXiv: Classical Analysis and ODEs, 2004
    Co-Authors: Yuval Peres, Boris Solomyak
    Abstract:

    In a recent paper, Pertti Mattila asked which gauge functions $\phi$ have the property that for any planar Borel set $A$ with positive Hausdorff Measure in gauge $\phi$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $\phi(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $\phi$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.

  • equivalence of positive Hausdorff Measure and the open set condition for self conformal sets
    Proceedings of the American Mathematical Society, 2001
    Co-Authors: Yuval Peres, Karoly Simon, Michal Rams, Boris Solomyak
    Abstract:

    A compact set K is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the "open set condition" (OSC), then K has positive sdimensional Hausdorff Measure, where s is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the s-dimensional Hausdorff Measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting.

  • self similar sets of zero Hausdorff Measure and positive packing Measure
    Israel Journal of Mathematics, 2000
    Co-Authors: Yuval Peres, Karoly Simon, Boris Solomyak
    Abstract:

    We prove that there exist self-similar sets of zero Hausdorff Measure, but positive and finite packing Measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑08an4−nan4−n with digits withan ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff Measure result using special properties of self-similar sets, but the result on packing Measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff Measure of an arbitrary Borel set to the packing Measure of its projections.

Yuval Peres - One of the best experts on this subject based on the ideXlab platform.

  • The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure
    Further Developments in Fractals and Related Fields, 2013
    Co-Authors: Yuval Peres, Boris Solomyak
    Abstract:

    In an earlier work, joint with R. Kenyon, we computed the Hausdorff dimension of the “multiplicative golden mean shift” defined as the set of all reals in [0, 1] whose binary expansion (x k ) satisfies x k x 2k = 0 for all k ≥ 1. Here we show that this set has infinite Hausdorff Measure in its dimension. A more precise result in terms of gauges in which the Hausdorff Measure is infinite is also obtained.

  • the sharp Hausdorff Measure condition for length of projections
    Proceedings of the American Mathematical Society, 2005
    Co-Authors: Yuval Peres, Boris Solomyak
    Abstract:

    In a recent paper, Pertti Mattila asked which gauge functions ' have the property that for any Borel set AR 2 with Hausdorff Measure H ' (A) > 0, the projection of A to almost every line has positive length. We show that finiteness of R 1 0 '(r) r2 dr, which is known to be sufficient for this property, is also necessary for regularly varying '. Our proof is based on a random construction adapted to the gauge function.

  • the sharp Hausdorff Measure condition for length of projections
    arXiv: Classical Analysis and ODEs, 2004
    Co-Authors: Yuval Peres, Boris Solomyak
    Abstract:

    In a recent paper, Pertti Mattila asked which gauge functions $\phi$ have the property that for any planar Borel set $A$ with positive Hausdorff Measure in gauge $\phi$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $\phi(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $\phi$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.

  • equivalence of positive Hausdorff Measure and the open set condition for self conformal sets
    Proceedings of the American Mathematical Society, 2001
    Co-Authors: Yuval Peres, Karoly Simon, Michal Rams, Boris Solomyak
    Abstract:

    A compact set K is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the "open set condition" (OSC), then K has positive sdimensional Hausdorff Measure, where s is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the s-dimensional Hausdorff Measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting.

  • self similar sets of zero Hausdorff Measure and positive packing Measure
    Israel Journal of Mathematics, 2000
    Co-Authors: Yuval Peres, Karoly Simon, Boris Solomyak
    Abstract:

    We prove that there exist self-similar sets of zero Hausdorff Measure, but positive and finite packing Measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑08an4−nan4−n with digits withan ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff Measure result using special properties of self-similar sets, but the result on packing Measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff Measure of an arbitrary Borel set to the packing Measure of its projections.

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