The information content of points on lines and $k$-plane extensions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911207714193408 |
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| author | Fiedler, Jacob B. |
| author_facet | Fiedler, Jacob B. |
| contents | We prove a new lower bound on the algorithmic information content of points lying on a line in $\mathbb{R}^n$. More precisely, we show that a typical point $z$ on any line $\ell$ satisfies
\begin{equation*}
K_r(z)\geq \frac{K_r(\ell)}{2} + r - o(r)
\end{equation*}
at every precision $r$. In other words, a randomly chosen point on a line has (at least) half of the complexity of the line plus the complexity of its first coordinate. We apply this effective result to establish a classical bound on how much the Hausdorff dimension of a union of positive measure subsets of $k$-planes can increase when each subset is replaced with the entire $k$-plane. To prove the complexity bound, we modify a recent idea of Cholak-Csörnyei-Lutz-Lutz-Mayordomo-Stull. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The information content of points on lines and $k$-plane extensions Fiedler, Jacob B. Classical Analysis and ODEs Logic 68Q30, 28A78, 28A80 We prove a new lower bound on the algorithmic information content of points lying on a line in $\mathbb{R}^n$. More precisely, we show that a typical point $z$ on any line $\ell$ satisfies \begin{equation*} K_r(z)\geq \frac{K_r(\ell)}{2} + r - o(r) \end{equation*} at every precision $r$. In other words, a randomly chosen point on a line has (at least) half of the complexity of the line plus the complexity of its first coordinate. We apply this effective result to establish a classical bound on how much the Hausdorff dimension of a union of positive measure subsets of $k$-planes can increase when each subset is replaced with the entire $k$-plane. To prove the complexity bound, we modify a recent idea of Cholak-Csörnyei-Lutz-Lutz-Mayordomo-Stull. |
| title | The information content of points on lines and $k$-plane extensions |
| topic | Classical Analysis and ODEs Logic 68Q30, 28A78, 28A80 |
| url | https://arxiv.org/abs/2510.11645 |