Algorithmic dimension of lattices

1992 ◽  
Vol 32 (3) ◽  
pp. 535-538
Author(s):  
V. D. Dzgoev
Keyword(s):  
Algorithmic Dimension

Models of Finite Algorithmic Dimension and Autostability

2000 ◽  
pp. 247-269
Author(s):  
Yuri L. Ershov ◽  
Sergei S. Goncharov
Keyword(s):  
Algorithmic Dimension

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

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.

Algorithmic dimension of nilpotent groups

1989 ◽  
Vol 30 (2) ◽  
pp. 210-217 ◽  
Author(s):  
S. S. Goncharov ◽  
B. N. Drobotun
Keyword(s):  
Nilpotent Groups ◽  
Algorithmic Dimension

Nilpotent groups of finite algorithmic dimension

1989 ◽  
Vol 30 (1) ◽  
pp. 63-68 ◽  
Author(s):  
S. S. Goncharov ◽  
A. V. Molokov ◽  
N. S. Romanovskii
Keyword(s):  
Nilpotent Groups ◽  
Algorithmic Dimension

scholarly journals Fractal Intersections and Products via Algorithmic Dimension

2021 ◽  
Vol 13 (3) ◽  
pp. 1-15
Author(s):  
Neil Lutz
Keyword(s):  
Kolmogorov Complexity ◽  
Fractal Dimensions ◽  
Euclidean Spaces ◽  
Cartesian Products ◽  
Information Density ◽  
Algorithmic Dimension ◽  
Packing Dimensions ◽  
Algorithmic Information ◽  
Analytic Sets

Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.

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