Large dimensional sets not containing a given angle

• Autorzy: Viktor Harangi
• Języki publikacji: EN

Abstrakty

EN

We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝn of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors) that has dimension c(α) log n. The main result of the paper concerns the case of the angles π/3 and 2π/3. We present self-similar sets in ℝn of Hausdorff dimension $c{{\sqrt[3]{n}} \mathord{\left/ {\vphantom {{\sqrt[3]{n}} {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}$ with the property that they do not contain the angles π/3 and 2π/3. The constructed sets avoid not only the given angle α but also a small neighbourhood of α.

Słowa kluczowe

EN

Hausdorff dimension; Self-similar sets; Sets without given angles

Kategorie tematyczne

• 28A78: Hausdorff and packing measures
• 28A80: Fractals

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 4
• Strony: 757-764

Daty

wydano

2011-08-01

online

2011-05-26

Twórcy

autor

Viktor Harangi

• Hungarian Academy of Sciences

Bibliografia

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• [3] Harangi V., Keleti T., Kiss G., Maga P., Máthé A., Mattila P., Strenner B., How large dimension guarantees a given angle?, preprint available at http://arxiv.org/abs/1101.1426
• [4] Johnson W.B., Lindenstrauss J., Extensions of Lipschitz mappings into a Hilbert space, In: Conference in Modern Analysis and Probability, New Haven, 1982, Contemp. Math., 26, American Mathematical Society, Providence, 1984, 189–206
• [5] Keleti T., Construction of one-dimensional subsets of the reals not containing similar copies of given patterns, Anal. PDE, 2008, 1(1), 29–33 http://dx.doi.org/10.2140/apde.2008.1.29
• [6] Maga P., Full dimensional sets without given patterns, Real Anal. Exchange, 2010, 36(1), 79–90
• [7] Salmon G., A Treatise on the Analytic Geometry of Three Dimensions, 2nd ed., Hodges, Smith, and Company, Cambridge, 1865

Identyfikatory

DOI

10.2478/s11533-011-0043-x