Projection Theorems for $Φ$-Intermediate Dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917411793403904 |
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| author | Daw, Lara Attia, Najmeddine |
| author_facet | Daw, Lara Attia, Najmeddine |
| contents | $Φ$-intermediate dimensions interpolate between Hausdorff and box-counting dimensions by restricting admissible coverings to scale windows of the form $[Φ(r),r]$. Using a family of $Φ$-dependent kernels, we develop a potential-theoretic framework that characterizes these dimensions in terms of capacities and leads to associated $Φ$-dimension profiles. This framework provides effective tools for obtaining lower bounds from uniform potential estimates. As an application, we prove Marstrand--Mattila type projection theorems, showing that for $γ_{n,m}$-almost all $m$-dimensional subspaces $V$, the $Φ$-intermediate dimensions of $π_V E$ coincide with deterministic profile values depending only on $E$ and $m$. We also discuss consequences for continuity at the Hausdorff end-point and for the box dimensions of typical projections. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14337 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Projection Theorems for $Φ$-Intermediate Dimensions Daw, Lara Attia, Najmeddine Metric Geometry Primary 28A80, Secondary 28A78, 28A75, 31C15 $Φ$-intermediate dimensions interpolate between Hausdorff and box-counting dimensions by restricting admissible coverings to scale windows of the form $[Φ(r),r]$. Using a family of $Φ$-dependent kernels, we develop a potential-theoretic framework that characterizes these dimensions in terms of capacities and leads to associated $Φ$-dimension profiles. This framework provides effective tools for obtaining lower bounds from uniform potential estimates. As an application, we prove Marstrand--Mattila type projection theorems, showing that for $γ_{n,m}$-almost all $m$-dimensional subspaces $V$, the $Φ$-intermediate dimensions of $π_V E$ coincide with deterministic profile values depending only on $E$ and $m$. We also discuss consequences for continuity at the Hausdorff end-point and for the box dimensions of typical projections. |
| title | Projection Theorems for $Φ$-Intermediate Dimensions |
| topic | Metric Geometry Primary 28A80, Secondary 28A78, 28A75, 31C15 |
| url | https://arxiv.org/abs/2604.14337 |