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Interpolating with generalized Assouad dimensions.

Amlan Banaji1, Alex Rutar1, Sascha Troscheit2

  • 1Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland.

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The study introduces phi-Assouad dimensions, interpolating between upper box and Assouad dimensions. These dimensions reveal scale sensitivity and phase transitions in sets, with new properties and formulas established for various fractal sets.

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Area of Science:

  • Fractal Geometry
  • Metric Space Theory
  • Measure Theory

Background:

  • The Assouad spectrum and Assouad dimension are key fractal geometry concepts.
  • Understanding scale sensitivity in fractal sets is crucial for their characterization.
  • Interpolating dimensions can provide finer insights into fractal structures.

Purpose of the Study:

  • To introduce and establish key properties of the phi-Assouad dimensions.
  • To demonstrate the relationship between phi-Assouad dimensions and other fractal dimensions.
  • To apply these dimensions to analyze specific fractal constructions like Galton-Watson trees and self-similar sets.

Main Methods:

  • Utilizing properties of bounded doubling metric spaces.
  • Applying large deviations theorems for Galton-Watson processes.
  • Developing a general Borel-Cantelli-type lemma for random tree structures.
  • Analyzing overlapping self-similar sets and sequences with decreasing gaps.

Main Results:

  • Proving the existence of a phi-Assouad dimension equal to a given alpha for suitable metric spaces.
  • Demonstrating that the upper dimension is determined by the phi-Assouad dimension.
  • Deriving a precise formula for phi-Assouad dimensions of Galton-Watson tree boundaries.
  • Establishing results for overlapping self-similar sets and specific sequences.

Conclusions:

  • The phi-Assouad dimensions offer a generalized framework for studying fractal scale sensitivity.
  • These dimensions provide a more nuanced understanding of fractal behavior, particularly near phase transitions.
  • The study provides powerful tools for analyzing complex fractal sets arising in probability and geometry.