Large dimensional sets not containing a given angle

Viktor Harangi

Open Mathematics (2011)

  • Volume: 9, Issue: 4, page 757-764
  • ISSN: 2391-5455

Abstract

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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 n 3 n 3 log n 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.

How to cite

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Viktor Harangi. "Large dimensional sets not containing a given angle." Open Mathematics 9.4 (2011): 757-764. <http://eudml.org/doc/268964>.

@article{ViktorHarangi2011,
abstract = {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 $c\{\{\@root 3 \of \{n\}\} \mathord \{\left\bad. \{\vphantom\{\{\@root 3 \of \{n\}\} \{\log n\}\}\} \right. \hspace\{0.0pt\}\} \{\log n\}\}$ 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.},
author = {Viktor Harangi},
journal = {Open Mathematics},
keywords = {Hausdorff dimension; Self-similar sets; Sets without given angles; self-similar sets; sets without given angles},
language = {eng},
number = {4},
pages = {757-764},
title = {Large dimensional sets not containing a given angle},
url = {http://eudml.org/doc/268964},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Viktor Harangi
TI - Large dimensional sets not containing a given angle
JO - Open Mathematics
PY - 2011
VL - 9
IS - 4
SP - 757
EP - 764
AB - 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 $c{{\@root 3 \of {n}} \mathord {\left\bad. {\vphantom{{\@root 3 \of {n}} {\log n}}} \right. \hspace{0.0pt}} {\log n}}$ 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.
LA - eng
KW - Hausdorff dimension; Self-similar sets; Sets without given angles; self-similar sets; sets without given angles
UR - http://eudml.org/doc/268964
ER -

References

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  1. [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 Zbl0534.52007
  2. [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. [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 Zbl1279.28009
  4. [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 Zbl0539.46017
  5. [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 Zbl1151.28302
  6. [6] Maga P., Full dimensional sets without given patterns, Real Anal. Exchange, 2010, 36(1), 79–90 Zbl1246.28005
  7. [7] Salmon G., A Treatise on the Analytic Geometry of Three Dimensions, 2nd ed., Hodges, Smith, and Company, Cambridge, 1865 

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