Large dimensional sets not containing a given angle
Viktor Harangi
Open Mathematics, Tome 9 (2011), p. 757-764 / Harvested from The Polish Digital Mathematics Library
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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 cn3n3lognlogn 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.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:268964 
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     author = {Viktor Harangi},
     title = {Large dimensional sets not containing a given angle},
     journal = {Open Mathematics},
     volume = {9},
     year = {2011},
     pages = {757-764},
     zbl = {1232.28008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0043-x}
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Viktor Harangi. Large dimensional sets not containing a given angle. Open Mathematics, Tome 9 (2011) pp. 757-764. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0043-x/

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