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Fractal dimensions of neighborhoods

Procedure

Fractal dimensions of neighborhoods were calculated for a more sensitive determination of the most enriched neighborhoods. A fractal dimension for a neighborhood of a center of radius $rt+1$ with t + 1 closest distinct points in it can be calculated as follows. Let each point in the neighborhood be exchanged with the same neighborhood compressed with the $minimumDistancemaximumDistance≅r12⋅rt+1$ ratio. Next, each point in the compressed neighborhood N1 will be exchanged with the double compressed original neighborhood: compressed with the $r12⋅rt+12$ ratio, next each point of the double compressed neighborhood will be exchanged with the triple compressed neighborhood: $r12⋅rt+13$ ratio, and so on.

In this nested construction, balls of diminishing radiuses cover the infinite sequence of the nested compressed self-similar (fractal) ball-neighborhoods. Hence, the Hausdorff fractal dimension of the ball-neighborhood can be calculated as a minimal dimension d of a ball (volume without compression is $rt+1d$) in order to get the lowest estimation for sum of volumes of infinite series of all diminishing balls that cover all points in this nested fractal construction.

The Hausdorff content of this series is

Dimension d of this Hausdorff content of the ball-neighborhood is defined as:

Alternatively, the equivalent formula will be as follows:

This infimum for dimension is calculated as:

Indeed, for the above limit under $n→∞$ to be equal to zero the inequality as follows has to be true:

Taking the logarithm from both parts, the inequality will be transformed into equivalent:

Taking infimum we will get:

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