Hausdorff dimension and uniform exponents in dimension two

Abstract : In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent µ ∈ (1/2, 1) is 2(1 − µ) when µ ≥ √ 2/2, whereas for µ < √ 2/2 it is greater than 2(1 − µ) and at most (3 − 2µ)(1 − µ)/(1 + µ + µ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when µ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for µ ≥ 0.565. .. .
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Submitted on : Friday, October 14, 2016 - 2:44:33 PM
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  • HAL Id : hal-01381425, version 1
  • ARXIV : 1610.06374

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Yann Bugeaud, Yitwah Cheung, Nicolas Chevallier. Hausdorff dimension and uniform exponents in dimension two. 2016. 〈hal-01381425〉

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