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ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

Péter Móra

Open publisher page 5 citations

Abstract

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

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What this paper is about

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

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OpenAlex reports 5 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

Key concepts: Hausdorff dimension, Hausdorff measure, Sierpinski triangle, Mathematics, Upper and lower bounds, Measure (data warehouse), Sierpinski carpet, Combinatorics

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