Hausdorff dimension of the set of almost convergent sequences

The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

Asymptotics of the Hausdorff dimensions of the Julia sets of McMullen maps with error bounds*

For McMullen maps f λ (z) = z p + λ/z p , where λ ∈ C \ { 0 } , it is known that if p ⩾ 3 and λ is small enough, then the Julia set J(f λ ) of f λ is a Cantor set of circles. In this paper we show that the Hausdorff dimension of J(f λ ) has the following asymptotic behavior dim H J ( f λ ) = 1 + log 2 log p + O ( | λ | 2 − 4 / p ) , as λ → 0 . An explicit error estimation of the remainder is also obtained. We also observe a ‘dimension paradox’ for the Julia set of Cantor set of circles.

Dimension spectra of lines1

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.