The point-cross dimension
Abstract
Two transverse segments have local Hausdorff and upper box dimension one at their crossing, although they carry two independent tangent directions. We define the Point-Cross Dimension by assigning each tangent line the largest local upper box dimension captured by intersection with a regular C1 curve in that direction, then taking the supremum of their sums over finite independent families of tangent lines. Every Bouligand direction admits such a recurrent curve. The resulting profile is invariant under closure and covariant under local C1-diffeomorphisms. Each directional weight is attained, and the profile is upper semicontinuous. An integral representation by ranks of weight superlevel sets gives submodularity and, together with projection bounds, exact Cartesian-product additivity. Germs with the same projective tangent support, local upper box dimension, and rank of segment directions can have different Point-Cross Dimensions. Further computations give exact directional profiles for chirps and the Point-Cross Dimension of a self-affine carpet. Finally, approximation of a ray with error O(td), for d>1, forces directional weight at least 1−1/d, and this bound is sharp.
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BibTeX
@misc{maaroufi2026pointcross,
title = {{The point-cross dimension}},
author = {Nadir Maaroufi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08612},
eprint = {2607.08612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08612},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The point-cross dimension
AU - Nadir Maaroufi
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2607.08612
UR - https://arxiv.org/abs/2607.08612
AB - Two transverse segments have local Hausdorff and upper box dimension one at their crossing, although they carry two independent tangent directions. We define the Point-Cross Dimension by assigning each tangent line the largest local upper box dimension captured by intersection with a regular C^1 curve in that direction, then taking the supremum of their sums over finite independent families of tangent lines. Every Bouligand direction admits such a recurrent curve. The resulting profile is invariant under closure and covariant under local C^1-diffeomorphisms. Each directional weight is attained, and the profile is upper semicontinuous. An integral representation by ranks of weight superlevel sets gives submodularity and, together with projection bounds, exact Cartesian-product additivity. Germs with the same projective tangent support, local upper box dimension, and rank of segment directions can have different Point-Cross Dimensions. Further computations give exact directional profiles for chirps and the Point-Cross Dimension of a self-affine carpet. Finally, approximation of a ray with error O(t^d), for d>1, forces directional weight at least 1−1/d, and this bound is sharp.
ER -