Finitely stable fractal dimensions of compact sets are always matched by some convergent sequence inside the set, and dimension-homogeneous sets admit one sequence matching many dimensions simultaneously.
Edgar.Measure, Topology and Fractal Geometry, Springer-Verlag, New York, 1990
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The sequence property for fractal dimensions
Finitely stable fractal dimensions of compact sets are always matched by some convergent sequence inside the set, and dimension-homogeneous sets admit one sequence matching many dimensions simultaneously.