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Intersections of fractal sets with rectifiable curves

  • Oleg Ivrii
  • Truong Vu
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Abstract

Let K ⊂ℝd, d≥2, be a connected compact set and ϕ(t)=tL(t) be a gauge function with L non-increasing. We study intersections of K with rectifiable curves and chord-arc curves. We show that if ∫01dt/ϕ(t)<∞, then there exists a rectifiable curve γ⊂ℝd such that ℋϕ(K ∩γ)>0. Moreover, for every 0<s<1, there exists a chord-arc curve γ such that H.dim (K ∩γ)≥ s. Conversely, we show that if K is uniformly porous on lines and ∫01dt/ϕ(t)=∞, then ℋϕ(K ∩γ)=0 for every rectifiable curve γ and H.dim (K ∩γ)<1 for every chord-arc curve γ.

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Cite

BibTeXRIS
BibTeX
@misc{ivrii2026intersections,
  title = {{Intersections of fractal sets with rectifiable curves}},
  author = {Oleg Ivrii and Truong Vu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41510},
  eprint = {2610.41510},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41510},
}
RIS
TY  - UNPB
TI  - Intersections of fractal sets with rectifiable curves
AU  - Oleg Ivrii
AU  - Truong Vu
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41510
UR  - https://arxiv.org/abs/2610.41510
AB  - Let K ⊂ℝ^d, d≥2, be a connected compact set and ϕ(t)=tL(t) be a gauge function with L non-increasing. We study intersections of K with rectifiable curves and chord-arc curves. We show that if ∫_0^1dt/ϕ(t)<∞, then there exists a rectifiable curve γ⊂ℝ^d such that ℋ^ϕ(K ∩γ)>0. Moreover, for every 0<s<1, there exists a chord-arc curve γ such that H.dim (K ∩γ)≥ s. Conversely, we show that if K is uniformly porous on lines and ∫_0^1dt/ϕ(t)=∞, then ℋ^ϕ(K ∩γ)=0 for every rectifiable curve γ and H.dim (K ∩γ)<1 for every chord-arc curve γ.
ER  -