Related papers: Optimal Oracles for Point-to-Set Principles
Marstrand's celebrated projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. It is straightforward to see that sets for which the Fourier and…
In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost…
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…
We develop quantitative algorithmic information bounds for orthogonal projections and distances in the plane. Under mild independence conditions, the distance $|x-y|$ and a projection coordinate $p_e x$ each retain at least half the…
We prove that every real number in [0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals over the field of rationals. The logic of our proof is of particular interest. The statement of our theorem is classical; it…
Assuming $V=L$, we construct a plane set $E$ of Hausdorff dimension $1$ whose every orthogonal projection onto straight lines through the origin has Hausdorff dimension $0$. This is a counterexample to J. M. Marstrand's seminal projection…
We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a…
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…
Effective dimension has proven very useful in geometric measure theory through the point-to-set principle \cite{LuLu18}\ that characterizes Hausdorff dimension by relativized effective dimension. Finite-state dimension is the least…
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…
Marstrand's projection theorem from $1954$ states that if $K \subset \mathbb{R}^{3}$ is an analytic set, then, for $\mathcal{H}^{2}$ almost every $e \in S^{2}$, the orthogonal projection $\pi_{e}(K)$ of $K$ to the line spanned by $e$ has…
The macroscopic Hausdorff dimension Dim H (E) of a set E $\subset$ R d was introduced by Barlow and Taylor to quantify a "fractal at large scales" behavior of unbounded, possibly discrete, sets E. We develop a method based on potential…
Let $1 \leq m < s \leq n$ and let $A \subseteq \mathbb{R}^n$ be a Borel set of with $s$-dimensional Hausdorff measure $\mathcal{H}^s(A) > 0$. The classical Marstrand slicing theorem states that, for almost every $m$-dimensional subspace $V…
We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such…
In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the…
Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$…
This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We…
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems…
In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of $A \cdot A+...+A \cdot A$, where $A$ is a subset of the real line of a given Hausdorff…
For a fixed $\theta^2=1/m$, $m \in \mathbb{N}_+$, let $x \in [0, \theta)$ and $[a_1(x) \theta, a_2(x) \theta, \ldots]$ be the $\theta$-expansion of $x$. Our first goal is to extend for $\theta$-expansions the results of Jarnik \cite{J-1928}…