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Beatty sequences and multiplicative number theory

100%

Abercrombie A. G.

Acta Arithmetica

1995

tom 70

nr 3

195-207

On the Hausdorff dimension of certain self-affine sets

63%

Abercrombie A. G., Nair R.

Studia Mathematica

2002

tom 152

nr 2

105-124

A subset E of ℝⁿ is called self-affine with respect to a collection {ϕ₁,...,ϕₜ} of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ(S) = ⋃ _{1≤j≤t} ϕ_{j}(S)$. If Φ(S) ⊂ S let $E_{Φ}(S)$ denote $⋂ _{k≥0}{Φ}^k(S)$. For given Φ consisting of contracting "pseudo-dilations" (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ}(S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate in detail the special class of cases in ℝ², where the images of a fixed square under some maps ϕ₁,...,ϕₜ are some vertical and some horizontal rectangles of sides 1 and 2. This investigation is made possible by an extension of the method of calculating Hausdorff dimension developed by P. Billingsley.

Arithmetic functions on Beatty sequences

51%

Abercrombie A. G., Banks W. D., Shparlinski I. E.

Acta Arithmetica

2009

tom 136

nr 1

81-89

Ograniczanie wyników

2 Acta Arithmetica

1 Studia Mathematica

3 Abercrombie A. G.

1 Banks W. D.

1 Nair R.

1 Shparlinski I. E.

1 2009

1 2002

1 1995

Wyniki wyszukiwania

Beatty sequences and multiplicative number theory

On the Hausdorff dimension of certain self-affine sets

Arithmetic functions on Beatty sequences