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pro vyhledávání: '"Orponen P"'
This article is a study guide for ``On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" by Orponen and Shmerkin. We begin by introducing Furstenberg set problem and exceptional set of projections and provide a summ
Externí odkaz:
http://arxiv.org/abs/2406.04580
Autor:
Orponen, Tuomas, Ren, Kevin
We show that the "sharp Kaufman projection theorem" from 2023 is sharp in the class of Ahlfors $(1,\delta^{-\epsilon})$-regular sets. This is in contrast with a recent result of the first author, which improves the projection theorem in the class of
Externí odkaz:
http://arxiv.org/abs/2411.04528
Autor:
Orponen, Tuomas
This paper contains the following $\delta$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,\epsilon > 0$ and $s \in [0,1]$, there exists $\kappa > 0$ such that the following holds for all $\delta > 0$ small enough. Le
Externí odkaz:
http://arxiv.org/abs/2410.06872
Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.
Comment: 43 pages, 1 figure
Comment: 43 pages, 1 figure
Externí odkaz:
http://arxiv.org/abs/2407.00306
Autor:
Orponen, Tuomas, Yi, Guangzeng
Let $P \subset \mathbb{R}^{2}$ be a Katz-Tao $(\delta,s)$-set, and let $\mathcal{L}$ be a Katz-Tao $(\delta,t)$-set of lines in $\mathbb{R}^{2}$. A recent result of Fu and Ren gives a sharp upper bound for the $\delta$-covering number of the set of i
Externí odkaz:
http://arxiv.org/abs/2402.12104
Let $s \in [0,1]$ and $t \in [0,\min\{3s,s + 1\})$. Let $\sigma$ be a Borel measure supported on the parabola $\mathbb{P} = \{(x,x^{2}) : x \in [-1,1]\}$ satisfying the $s$-dimensional Frostman condition $\sigma(B(x,r)) \leq r^{s}$. Answering a quest
Externí odkaz:
http://arxiv.org/abs/2401.17867
Autor:
Orponen, Tuomas
Let $t \in (1,2)$, and let $B \subset \mathbb{R}^{2}$ be a Borel set with $\dim_{\mathrm{H}} B > t$. I show that $$\mathcal{H}^{1}(\{e \in S^{1} : \dim_{\mathrm{H}} (B \cap \ell_{x,e}) \geq t - 1\}) > 0$$ for all $x \in \mathbb{R}^{2} \, \setminus \,
Externí odkaz:
http://arxiv.org/abs/2311.14481
One formulation of Marstrand's slicing theorem is the following. Assume that $t \in (1,2]$, and $B \subset \mathbb{R}^{2}$ is a Borel set with $\mathcal{H}^{t}(B) < \infty$. Then, for almost all directions $e \in S^{1}$, $\mathcal{H}^{t}$ almost all
Externí odkaz:
http://arxiv.org/abs/2310.11219
We prove the following. Let $\mu_{1},\ldots,\mu_{n}$ be Borel probability measures on $[-1,1]$ such that $\mu_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative convolution o
Externí odkaz:
http://arxiv.org/abs/2309.03068
We present a general method for rendering representations of multi-stranded DNA complexes from textual descriptions into 2D diagrams. The complexes can be arbitrarily pseudoknotted, and if a planar rendering is possible, the method will determine one
Externí odkaz:
http://arxiv.org/abs/2308.06392