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2011 | 9 | 4 | 757-764
Tytuł artykułu

Large dimensional sets not containing a given angle

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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.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
4
Strony
757-764
Opis fizyczny
Daty
wydano
2011-08-01
online
2011-05-26
Twórcy
Bibliografia
  • [1] Erdős P., Füredi Z., The greatest angle among n points in the d-dimensional Euclidean space, In: Combinatorial Mathematics, Marseille-Luminy, 1981, North-Holland Math. Stud., 75, North-Holland, Amsterdam, 1983, 275–283
  • [2] Falconer K.J., On a problem of Erdős on fractal combinatorial geometry, J. Combin. Theory Ser. A, 1992, 59(1), 142–148 http://dx.doi.org/10.1016/0097-3165(92)90106-5
  • [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
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0043-x
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