Equilateral triangles in subsets of ℝd of large Hausdorff dimension

Springer Science and Business Media LLC - Tập 231 - Trang 123-137 - 2019
Alex Iosevich1, Bochen Liu1
1Department of Mathematics, University of Rochester, Rochester, USA

Tóm tắt

We prove that subsets of ℝd, d ≥ 4 of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two dimensions (see [3]). We show that no extra conditions are needed in dimensions four and higher. The three dimensional case remains open. Some interesting parallels exist between the triangle problem in Euclidean space and its counterpart in vector spaces over finite fields. We shall outline these similarities in hopes of eventually achieving a comprehensive understanding of this phenomenon in the setting of locally compact abelian groups.

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Springer Science and Business Media LLC

Tập 231

123-137

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