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Applications of dimension interpolation to orthogonal projections
1University of St Andrews, St Andrews, Scotland.
Research in the Mathematical Sciences
|March 11, 2025
Summary
Dimension interpolation unifies fractal dimension studies by exploring intermediate spectra. This approach reveals novel features and applications, particularly in the dimension theory of orthogonal projections.
Area of Science:
- Fractal Geometry
- Geometric Measure Theory
Background:
- Fractal dimension concepts like Hausdorff, box, Assouad, and Fourier dimensions are well-established.
- These individual dimensions may not capture all geometric features of complex sets.
Purpose of the Study:
- To introduce and survey the concept of dimension interpolation.
- To explore applications of dimension interpolation in the theory of orthogonal projections.
Main Methods:
- Reviewing existing literature on fractal dimensions.
- Analyzing dimension interpolation through various spectra (Fourier, intermediate, Assouad).
- Examining applications to orthogonal projections, building upon the Marstrand-Mattila theorem.
Main Results:
- Dimension interpolation offers a unified framework for studying fractal dimensions.
- Novel geometric insights are revealed by considering spectra between established dimensions.
- Specific applications demonstrate the power of interpolation in projection theory.
Conclusions:
- Dimension interpolation provides a richer understanding of fractal geometry.
- This framework enhances the study of orthogonal projections beyond traditional methods.
- Further research into dimension spectra promises new discoveries.
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