# Large dimensional sets not containing a given angle

Open Mathematics (2011)

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

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topViktor 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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- [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] 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] 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] Maga P., Full dimensional sets without given patterns, Real Anal. Exchange, 2010, 36(1), 79–90 Zbl1246.28005
- [7] Salmon G., A Treatise on the Analytic Geometry of Three Dimensions, 2nd ed., Hodges, Smith, and Company, Cambridge, 1865

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