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Jacob B. Fiedler

Publications and source records attributed to Jacob B. Fiedler.

8 recordsLinked to original sources
arXiv2026

Projections of sets with optimal oracles onto $k$-planes

We prove a Kaufman-type exceptional set estimate for sets in $\mathbb{R}^n$ that have optimal oracles, a class of sets that strictly contains the analytic sets and sets with equal Hausdorff and packing dimension. As a consequence, we generalize the conditions under which Marstrand's projection theorem for $k$-planes is known to hold. Our proofs use effective methods, especially Kolmogorov complexity, and along the way, we introduce several new tools for studying the information content of elements of the Grassmannian.

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arXiv2025

The information content of points on lines and $k$-plane extensions

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.

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arXiv2025

On the packing dimension of unions and extensions of $k$-planes

We study the packing dimension of unions of subsets of $k$-planes in $\mathbb{R}^n$ using tools from algorithmic information theory, obtaining an analog of a result of Héra and a mild generalization of a recent result of Fraser. Along the way, we introduce a notion of effective dimension on the Grassmannian and affine Grassmannian, and we establish several useful algorithmic and geometric tools in this setting. Additionally, we consider how the packing dimension of the union of certain subsets of $k$-planes changes when the subsets are extended to the entire $k$-plane. Finally, we improve the above bounds for unions and extensions in the special case that $k=n-1$.

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arXiv2025

Pinned distances of planar sets with low dimension

In this paper, we give improved bounds on the Hausdorff dimension of pinned distance sets of planar sets with dimension strictly less than one. As the planar set becomes more regular (i.e., the Hausdorff and packing dimension become closer), our lower bound on the Hausdorff dimension of the pinned distance set improves. Additionally, we prove the existence of small universal sets for pinned distances. In particular, we show that, if a Borel set $X\subseteq\mathbb{R}^2$ is weakly regular ($\dim_H(X) = \dim_P(X)$), and $\dim_H(X) > 1$, then \begin{equation*} \sup\limits_{x\in X}\dim_H(Δ_x Y) = \min\{\dim_H(Y), 1\} \end{equation*} for every Borel set $Y\subseteq\mathbb{R}^2$. Furthermore, if $X$ is also compact and Ahlfors-David regular, then for every Borel set $Y\subseteq\mathbb{R}^2$, there exists some $x\in X$ such that \begin{equation*} \dim_H(Δ_x Y) = \min\{\dim_H(Y), 1\}. \end{equation*}

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arXiv2025

Universal Sets for Projections

We investigate variants of Marstrand's projection theorem that hold for sets of directions and classes of sets in $\mathbb{R}^2$. We say that a set of directions $D \subseteq\mathcal{S}^1$ is $\textit{universal}$ for a class of sets if, for every set $E$ in the class, there is a direction $e\in D$ such that the projection of $E$ in the direction $e$ has maximal Hausdorff dimension. We construct small universal sets for certain classes. Particular attention is paid to the role of regularity. We prove the existence of universal sets with arbitrarily small positive Hausdorff dimension for the class of weakly regular sets. We prove that there is a universal set of zero Hausdorff dimension for the class of AD-regular sets.

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arXiv2025

Bounds on the dimension of lineal extensions

Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq \mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti's line segment extension conjecture posits that the Hausdorff dimension of $F$ should equal that of $E$. Working in $\mathbb{R}^2$, we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems.

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arXiv2023

Dimension of Pinned Distance Sets for Semi-Regular Sets

We prove that if $E\subseteq \R^2$ is analytic and $1<d < \dim_H(E)$, there are ``many'' points $x\in E$ such that the Hausdorff dimension of the pinned distance set $Δ_x E$ is at least $d\left(1 - \frac{\left(D-1\right)\left(D-d\right)}{2D^2+\left(2-4d\right)D+d^2+d-2}\right)$, where $D = \dim_P(E)$. In particular, we prove that $\dim_H(Δ_x E) \geq \frac{d(d-4)}{d-5}$ for these $x$, which gives the best known lower bound for this problem when $d \in (1, 5-\sqrt{15})$. We also prove that there exists some $x\in E$ such that the packing dimension of $Δ_x E$ is at least $\frac{12 -\sqrt{2}}{8\sqrt{2}}$. Moreover, whenever the packing dimension of $E$ is sufficiently close to the Hausdorff dimension of $E$, we show the pinned distance set $Δ_x E$ has full Hausdorff dimension for many points $x\in E$; in particular the condition is that $D<\frac{(3+\sqrt{5})d-1-\sqrt{5}}{2}$. We also consider the pinned distance problem between two sets $X, Y\subseteq \R^2$, both of Hausdorff dimension greater than 1. We show that if either $X$ or $Y$ has equal Hausdorff and packing dimensions, the pinned distance $Δ_x Y$ has full Hausdorff dimension for many points $x\in X$.

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