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Tuomas Orponen

Publications and source records attributed to Tuomas Orponen.

At least 37 records · Page 2Linked to original sources
arXiv2023

Boundedness of singular integrals on $C^{1,α}$ intrinsic graphs in the Heisenberg group

We study singular integral operators induced by $3$-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is $L^{2}$ bounded on vertical planes, with uniform constants, then it is also $L^{2}$ bounded on all intrinsic graphs of compactly supported $C^{1,α}$ functions over vertical planes. In particular, the result applies to the operator $\mathcal{R}$ induced by the kernel $$\mathcal{K}(z) = \nabla_{\mathbb{H}} \| z \|^{-2}, \quad z \in \mathbb{H} \setminus \{0\},$$ the horizontal gradient of the fundamental solution of the sub-Laplacian. The $L^{2}$ boundedness of $\mathcal{R}$ is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of non-removable sets with positive and locally finite $3$-dimensional measure.

math.CA↗
arXiv2023

On the Hausdorff dimension of radial slices

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 $x \in \mathbb{R}^{2} \, \setminus \, E$, where $\dim_{\mathrm{H}} E \leq 2 - t$. This is the sharp bound for $\dim_{\mathrm{H}} E$. The main technical tool is an incidence inequality of the form $$\mathcal{I}_δ(μ,ν) \lesssim_{t} δ\cdot \sqrt{I_{t}(μ)I_{3 - t}(ν)}, \qquad t \in (1,2),$$ where $μ$ is a Borel measure on $\mathbb{R}^{2}$, and $ν$ is a Borel measure on the set of lines in $\mathbb{R}^{2}$, and $\mathcal{I}_δ(μ,ν)$ measures the $δ$-incidences between $μ$ and the lines parametrised by $ν$. This inequality can be viewed as a $δ^{-ε}$-free version of a recent incidence theorem due to Fu and Ren. The proof in this paper avoids the high-low method, and the induction-on-scales scheme responsible for the $δ^{-ε}$-factor in Fu and Ren's work. Instead, the inequality is deduced from the classical smoothing properties of the $X$-ray transform.

math.CA↗
arXiv2023

On the discretised $ABC$ sum-product problem

Let $0 < β\leq α< 1$ and $κ> 0$. I prove that there exists $η> 0$ such that the following holds for every pair of Borel sets $A,B \subset \mathbb{R}$ with $\dim_{\mathrm{H}} A = α$ and $\dim_{\mathrm{H}} B = β$: $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq α+ η\} \leq \tfrac{α- β}{1 - β} + κ.$$ This extends a result of Bourgain from 2010, which contained the case $α= β$. The paper also contains a $δ$-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form $a_{1}B + \ldots + a_{n}B$.

math.CO↗
arXiv2023

How much can heavy lines cover?

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 of $B$ is covered by lines $\ell$ parallel to $e$ with $\dim_{\mathrm{H}} (B \cap \ell) = t - 1$. We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction $e \in S^{1}$, is it true that a strictly less than $t$-dimensional part of $B$ is covered by the heavy lines $\ell \subset \mathbb{R}^{2}$, namely those with $\dim_{\mathrm{H}} (B \cap \ell) > t - 1$? A positive answer for $t$-regular sets $B \subset \mathbb{R}^{2}$ was previously obtained by the first author. The answer for general Borel sets turns out to be negative for $t \in (1,\tfrac{3}{2}]$ and positive for $t \in (\tfrac{3}{2},2]$. More precisely, the heavy lines can cover up to a $\min\{t,3 - t\}$ dimensional part of $B$ in a generic direction. We also consider the part of $B$ covered by the $s$-heavy lines, namely those with $\dim_{\mathrm{H}} (B \cap \ell) \geq s$ for $s > t - 1$. We establish a sharp answer to the question: how much can the $s$-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors-regular sets to the class of sub-uniformly distributed sets.

math.CA↗
arXiv2023

On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

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 $\dim_{\mathrm{H}} \mathcal{L} \geq t$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$, and $t \in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in $\mathbb{R}^{2}$ is no smaller than $2s + ε$, where $ε> 0$ depends only on $s$ and $t$. For $s>1/2$ and $t = 1$, this is an $ε$-improvement over a result of Wolff from 1999. The same method also yields an $ε$-improvement to Kaufman's projection theorem from 1968. We show that if $s \in (0,1)$, $t \in (s,2]$ and $K \subset \mathbb{R}^{2}$ is an analytic set with $\dim_{\mathrm{H}} K = t$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{H}} π_{e}(K) \leq s\} \leq s - ε,$$ where $ε> 0$ only depends on $s$ and $t$. Here $π_{e}$ is the orthogonal projection to $\mathrm{span}(e)$.

math.CA↗
arXiv2023

Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences

Let $\{π_{e} \colon \mathbb{H} \to \mathbb{W}_{e} : e \in S^{1}\}$ be the family of vertical projections in the first Heisenberg group $\mathbb{H}$. We prove that if $K \subset \mathbb{H}$ is a Borel set with Hausdorff dimension $\dim_{\mathbb{H}} K \in [0,2] \cup \{3\}$, then $$ \dim_{\mathbb{H}} π_{e}(K) \geq \dim_{\mathbb{H}} K $$ for $\mathcal{H}^{1}$ almost every $e \in S^{1}$. This was known earlier if $\dim_{\mathbb{H}} K \in [0,1]$. The proofs for $\dim_{\mathbb{H}} K \in [0,2]$ and $\dim_{\mathbb{H}} K = 3$ are based on different techniques. For $\dim_{\mathbb{H}} K \in [0,2]$, we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl. To handle the case $\dim_{\mathbb{H}} K = 3$, we introduce a point-line duality between horizontal lines and conical lines in $\mathbb{R}^{3}$. This allows us to transform the Heisenberg problem into a point-plate incidence question in $\mathbb{R}^{3}$. To solve the latter, we apply a Kakeya inequality for plates in $\mathbb{R}^{3}$, due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets $K \subset \mathbb{H}$ with $\dim_{\mathbb{H}} K \in (5/2,3)$.

math.CA↗
arXiv2023

Cheeger's differentiation theorem via the multilinear Kakeya inequality

Suppose that $(X,d,μ)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincaré inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbourhoods of Lipschitz graphs to show that any non-trivial measure with $n$ independent Alberti representations has Hausdorff dimension at least $n$.

math.MG↗
arXiv2022

A note on Kakeya sets of horizontal and $SL(2)$ lines

We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup \mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and Zahl. The $SL(2)$ lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal and conical lines in $\mathbb{R}^{3}$, combined with recent work on restricted families of projections to planes, due to Gan, Guo, Guth, Harris, Maldague, and Wang. Our result also has a corollary for Nikodym sets associated with horizontal lines, which answers a special case of a question of Kim.

math.CA↗
arXiv2022

Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let $X,Y \subset \mathbb{R}^{2}$ be non-empty Borel sets. If $X$ is not contained on any line, we prove that \[ \sup_{x \in X} \dim_{\mathrm{H}} π_{x}(Y) \geq \min\{\dim_{\mathrm{H}} X,\dim_{\mathrm{H}} Y,1\}. \] If $\dim_{\mathrm{H}} Y > 1$, we have the following improved lower bound: \[ \sup_{x \in X} \dim_{\mathrm{H}} π_{x}(Y \, \setminus \, \{x\}) \geq \min\{\dim_{\mathrm{H}} X + \dim_{\mathrm{H}} Y - 1,1\}. \] Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if $X \subset \mathbb{R}^{2}$ is a Borel set with the property that $\dim_{\mathrm{H}} (X \, \setminus \, \ell) = \dim_{\mathrm{H}} X$ for all lines $\ell \subset \mathbb{R}^{2}$, then the line set spanned by $X$ has Hausdorff dimension at least $\min\{2\dim_{\mathrm{H}} X,2\}$. While the results above concern $\mathbb{R}^{2}$, we also derive some counterparts in $\mathbb{R}^{d}$ by means of integralgeometric considerations. The proofs are based on an $ε$-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren.

math.CA↗
arXiv2022

Integrability of orthogonal projections, and applications to Furstenberg sets

Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\mathbb{R}^{d}$, and let $π_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $μ$ is a compactly supported Radon measure on $\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $μ(B(x,r)) \leq Cr^{s}$ for all $x \in \mathbb{R}^{d}$ and $r > 0$, then $$\int_{\mathcal{G}(d,n)} \|π_{V}μ\|_{L^{p}(V)}^{p} \, dγ_{d,n}(V) < \infty, \qquad 1 \leq p < \frac{2d - n - s}{d - s}.$$ The upper bound for $p$ is sharp, at least, for $d - 1 \leq s \leq d$, and every $0 < n < d$. Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of $(s,t)$-Furstenberg sets. For $0 \leq s \leq 1$ and $0 \leq t \leq 2$, a set $K \subset \mathbb{R}^{2}$ is called an $(s,t)$-Furstenberg set if there exists a $t$-dimensional family $\mathcal{L}$ of affine lines in $\mathbb{R}^{2}$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. As a consequence of our projection theorem in $\mathbb{R}^{2}$, we show that every $(s,t)$-Furstenberg set $K \subset \mathbb{R}^{2}$ with $1 < t \leq 2$ satisfies $$\dim_{\mathrm{H}} K \geq 2s + (1 - s)(t - 1).$$ This improves on previous bounds for pairs $(s,t)$ with $s > \tfrac{1}{2}$ and $t \geq 1 + ε$ for a small absolute constant $ε> 0$. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in $\mathbb{R}^{d}$. As another corollary of our method, we obtain a $δ$-discretised sum-product estimate for $(δ,s)$-sets. Our bound improves on a previous estimate of Chen for every $\tfrac{1}{2} < s < 1$, and also of Guth-Katz-Zahl for $s \geq 0.5151$.

math.CA↗
arXiv2022

On exceptional sets of radial projections

We prove two new exceptional set estimates for radial projections in the plane. If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K > 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} π_{x}(K) \leq σ\} \leq \max\{1 + σ- \dim_{\mathrm{H}} K,0\}, \qquad σ\in [0,1).$$ If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K \leq 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} π_{x}(K) < \dim_{\mathrm{H}} K\} \leq 1.$$ The finite field counterparts of both results above were recently proven by Lund, Thang, and Huong Thu. Our results resolve the planar cases of conjectures of Lund-Thang-Huong Thu, and Liu.

math.CA↗
arXiv2022

Additive properties of fractal sets on the parabola

Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K + K) \geq 2s + ε, $$ where $ε= ε(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $μ$ is a Borel measure on $\mathbb{P}$ satisfying $μ(B(x,r)) \leq r^{s}$ for all $x \in \mathbb{P}$ and $r > 0$, then there exists $ε= ε(s) > 0$ such that $$ \|\hatμ\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + ε)} $$ for all sufficiently large $R \geq 1$. The proof is based on a reduction to a $δ$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem.

math.CA↗
arXiv2022

Structure of sets with nearly maximal Favard length

Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see that $\mathrm{Fav}(E) \leq \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$ \mathrm{Fav}(E) \geq \mathrm{Fav}(L) - δ, $$ the set $E$ can be covered by an $ε$-Lipschitz graph, up to a set of length $ε$. The dependence between $ε$ and $δ$ is polynomial: in fact, the conclusions hold with $ε= Cδ^{1/70}$ for an absolute constant $C > 0$.

math.CA↗
arXiv2022

Hausdorff dimension bounds for the ABC sum-product problem

The purpose of this paper is to complete the proof of the following result. Let $0 < β\leq α< 1$ and $κ> 0$. Then, there exists $η> 0$ such that whenever $A,B \subset \mathbb{R}$ are Borel sets with $\dim_{\mathrm{H}} A = α$ and $\dim_{\mathrm{H}} B = β$, then $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq α+ η\} \leq \tfrac{α- β}{1 - β} + κ.$$ This extends a result of Bourgain from 2010, which contained the case $α= β$. This paper is a sequel to the author's previous work from 2021 which, roughly speaking, established the same result with $\dim_{\mathrm{H}} (A + cB)$ replaced by $\dim_{\mathrm{B}}(A + cB)$, the box dimension of $A + cB$. It turns out that, at the level of $δ$-discretised statements, the superficially weaker box dimension result formally implies the Hausdorff dimension result.

math.CA↗
arXiv2021

On arithmetic sums of Ahlfors-regular sets

Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: α$ and $\dim_{\mathrm{H}} B =: β$. I prove that $$\dim_{\mathrm{H}} [A + θB] \geq α+ β\cdot \tfrac{1 - α}{2 - α}$$ for all $θ\in \mathbb{R} \, \setminus \, E$, where $\dim_{\mathrm{H}} E = 0$.

math.CA↗
arXiv2021

Metric rectifiability of $\mathbb{H}$-regular surfaces with Hölder continuous horizontal normal

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole-Pauls, Bigolin-Vittone, and Antonelli-Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of $\mathbb{H}$-regular surfaces. We prove that $\mathbb{H}$-regular surfaces in $\mathbb{H}^{n}$ with $α$-Hölder continuous horizontal normal, $α> 0$, are metric bilipschitz rectifiable. This improves on the work by Antonelli-Le Donne, where the same conclusion was obtained for $C^{\infty}$-surfaces. In $\mathbb{H}^{1}$, we prove a slightly stronger result: every codimension-$1$ intrinsic Lipschitz graph with an $ε$ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between "big pieces" of metric spaces.

math.CA↗
arXiv2021

A Marstrand-type restricted projection theorem in $\mathbb{R}^{3}$

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 $π_{e}(K)$ of $K$ to the line spanned by $e$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. This paper contains the following sharper version of Marstrand's theorem. Let $V \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. Then, for $\mathcal{H}^{1}$ almost every $e \in S^{2} \cap V$, the projection $π_{e}(K)$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. For $0 \leq t < \dim_{\mathrm{H}} K$, we also prove an upper bound for the Hausdorff dimension of those vectors $e \in S^{2} \cap V$ with $\dim_{\mathrm{H}} ρ_{e}(K) \leq t < \dim_{\mathrm{H}} K$.

math.CA↗