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We construct a d Hausdorff dimensional compact set in Rd that does not contain the vertices of any parallelogram. We also prove that for any given triangle (3 given points in the plane) there exists a compact set in R2 of Hausdorff dimension 2 that does not contain any similar copy of the triangle. On the other hand, we show that the set of the 3-point(More)
We study the following two problems: (1) Given n ≥ 2 and α, how large Hausdorff dimension can a compact set A ⊂ R have if A does not contain three points that form an angle α? (2) Given α and δ, how large Hausdorff dimension can a compact subset A of a Euclidean space have if A does not contain three points that form an angle in the δ-neighborhood of α?(More)
We study the following two problems: (1) Given n ≥ 2 and α, how large Hausdorff dimension can a compact set A ⊂ R have if A does not contain three points that form an angle α? (2) Given α and δ, how large Hausdorff dimension can a compact subset A of a Euclidean space have if A does not contain three points that form an angle in the δ-neighborhood of α?(More)
Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such(More)
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