Bibliographic Information: Harris, T.L.J. (2024). Exceptional sets for length under restricted families of projections onto lines in R3. arXiv preprint arXiv:2408.04885v3.
Research Objective: This paper investigates the size of exceptional sets, specifically focusing on the Hausdorff dimension of sets of projections of Borel sets in R3 onto lines that result in zero length.
Methodology: The author utilizes a "small cap" wave packet decomposition for the cone, a technique inspired by previous work on the distance set problem and restricted projection problems. This method allows for a more refined analysis of the "bad" and "good" parts of the measure associated with the Borel set. The "good" part is further analyzed using a standard wave packet decomposition and a pigeonholing argument to optimize the bounds.
Key Findings: The paper establishes that if a Borel set in R3 has Hausdorff dimension greater than 1, the set of projections onto lines with zero length also has a bounded Hausdorff dimension. This bound is expressed as (3 - dimension of the original set) / 2, improving upon previous bounds.
Main Conclusions: The results refine existing exceptional set inequalities for projections of Borel sets in R3 onto lines. The study demonstrates a novel approach using a combination of small cap and standard wave packet decompositions, potentially generalizable to higher dimensions.
Significance: This work contributes to the field of geometric measure theory, specifically to the study of projections and Hausdorff dimension. The improved bounds on exceptional sets provide a deeper understanding of the geometric properties of projections.
Limitations and Future Research: The paper focuses on projections onto lines in R3. Exploring similar results for projections onto higher dimensional subspaces in higher dimensional Euclidean spaces could be a potential direction for future research. Additionally, investigating the sharpness of the obtained bounds would be of interest.
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by Terence L. J... at arxiv.org 10-15-2024
https://arxiv.org/pdf/2408.04885.pdfDeeper Inquiries