arXiv subjects

Tuomas Orponen

Publications and source records attributed to Tuomas Orponen.

At least 73 records · Page 4Linked to original sources

arXiv2018

Improved bounds for restricted families of projections to planes in $\mathbb{R}^{3}$

For $e \in S^{2}$, the unit sphere in $\mathbb{R}^3$, let $π_{e}$ be the orthogonal projection to $e^{\perp} \subset \mathbb{R}^{3}$, and let $W \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K \subset \mathbb{R}^{3}$ is a Borel set with $\dim_{\mathrm{H}} K \leq \tfrac{3}{2}$, then $\dim_{\mathrm{H}} π_{e}(K) = \dim_{\mathrm{H}} K$ for $\mathcal{H}^{1}$ almost every $e \in S^{2} \cap W$, where $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure and $\dim_{\mathrm{H}}$ the Hausdorff dimension. This was known earlier, due to Järvenpää, Järvenpää, Ledrappier and Leikas, for Borel sets $K \subset \mathbb{R}^{3}$ with $\dim_{\mathrm{H}} K \leq 1$. We also prove a partial result for sets with dimension exceeding $3/2$, improving earlier bounds by D. Oberlin and R. Oberlin.

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arXiv2018

The local symmetry condition in the Heisenberg group

I propose an analogue in the first Heisenberg group $\mathbb{H}$ of David and Semmes' local symmetry condition (LSC). For closed $3$-regular sets $E \subset \mathbb{H}$, I show that the (LSC) is implied by the $L^{2}(\mathcal{H}^{3}|_{E})$ boundedness of $3$-dimensional singular integrals with horizontally antisymmetric kernels, and that the (LSC) implies the weak geometric lemma for vertical $β$-numbers.

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arXiv2018

Intrinsic Lipschitz graphs and vertical $β$-numbers in the Heisenberg group

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $\mathbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical $β$-numbers. Conversely, extending a result of David and Semmes from $\mathbb{R}^{n}$, we prove that a $3$-Ahlfors-David regular subset in $\mathbb{H}$, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.

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arXiv2018

On the conformal dimension of product measures

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if $(X,d)$ is a metric space and $f \colon K_{E} \to (X,d)$ is a quasisymmetric homeomorphism, then $$\dim_{\mathrm{H}} f(K_{E}) \geq \dim_{\mathrm{H}} K_{E}.$$ We prove that the measure-theoretic analogue of the result is not true. For any $d \geq 2$ and $0 \leq s < d - 1$, there exist compact sets $E \subset \mathbb{R}^{d - 1}$ with $0 < \mathcal{H}^{s}(E) < \infty$ such that the conformal dimension of $ν$, the restriction of the $(1 + s)$-dimensional Hausdorff measure on $K_{E}$, is zero. More precisely, for any $ε> 0$, there exists a quasisymmetric embedding $F \colon K_{E} \to \mathbb{R}^{d}$ such that $\dim_{\mathrm{H}} F_{\sharp}ν< ε$.

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arXiv2018

A note on expansion in prime fields

Let $β,ε\in (0,1]$, and $k \geq \exp(122 \max\{1/β,1/ε\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^β$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, \qquad a_{1},\ldots,a_{k} \in A,$$ with $|S| \geq 2^{-12}p^{-ε}\min\{|A||B|,p\}$. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $α< 1$ and $β,δ> 0$, there exists $ε> 0$ such that the following holds. If $A,B,E \subset \mathbb{Z}_{p}$ satisfy $|A| \leq p^α$, $|B| \geq p^β$, and $|B||E| \geq p^δ|A|$, then there exists $t \in E$ such that $$|A + tB| \geq c p^ε|A|,$$ for some absolute constant $c > 0$. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.

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arXiv2018

Absolute continuity and $α$-numbers on the real line

Let $μ,ν$ be Radon measures on $\mathbb{R}$, with $μ$ non-atomic and $ν$ doubling, and write $μ= μ_{a} + μ_{s}$ for the Lebesgue decomposition of $μ$ relative to $ν$. For an interval $I \subset \mathbb{R}$, define $α_{μ,ν}(I) := \mathbb{W}_{1}(μ_{I},ν_{I})$, the Wasserstein distance of normalised blow-ups of $μ$ and $ν$ restricted to $I$. Let $\mathcal{S}_ν$ be the square function $$\mathcal{S}^{2}_ν(μ) = \sum_{I \in \mathcal{D}} α_{μ,ν}^{2}(I)χ_{I},$$ where $\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I prove that $\mathcal{S}_ν(μ)$ is finite $μ_{a}$ almost everywhere, and infinite $μ_{s}$ almost everywhere. I also prove a version of the result for a non-dyadic variant of the square function $\mathcal{S}_ν(μ)$. The results answer the simplest "$n = d = 1"$ case of a problem of J. Azzam, G. David and T. Toro.

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arXiv2017

An improved bound on the packing dimension of Furstenberg sets in the plane

Let $0 \leq s \leq 1$. A set $K \subset \mathbb{R}^{2}$ is a Furstenberg $s$-set, if for every unit vector $e \in S^{1}$, some line $L_{e}$ parallel to $e$ satisfies $$\dim_{\mathrm{H}} [K \cap L_{e}] \geq s.$$ The Furstenberg set problem, introduced by T. Wolff in 1999, asks for the best lower bound for the dimension of Furstenberg $s$-sets. Wolff proved that $\dim_{\mathrm{H}} K \geq \max\{s + 1/2,2s\}$ and conjectured that $\dim_{\mathrm{H}} K \geq (1 + 3s)/2$. The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that $\dim_{\mathrm{H}} K \geq 1 + ε$ for Furstenberg $1/2$-sets $K$, where $ε> 0$ is an absolute constant. In the present paper, I prove a similar $ε$-improvement for all $1/2 < s < 1$, but only for packing dimension: $\dim_{\mathrm{p}} K \geq 2s + ε$ for all Furstenberg $s$-sets $K \subset \mathbb{R}^{2}$, where $ε> 0$ only depends on $s$. The proof rests on a new incidence theorem for finite collections of planar points and tubes of width $δ> 0$. As another corollary of this theorem, I obtain a small improvement for Kaufman's estimate from 1968 on the dimension of exceptional sets of orthogonal projections. Namely, I prove that if $K \subset \mathbb{R}^{2}$ is a linearly measurable set with positive length, and $1/2 < s < 1$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{p}} π_{e}(K) \leq s\} \leq s - ε$$ for some $ε> 0$ depending only on $s$. Here $π_{e}$ is the orthogonal projection onto the line spanned by $e$.

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arXiv2017

On the absolute continuity of radial projections

Let $d \geq 2$ and $d - 1 < s < d$. Let $μ$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections $π_{x\sharp}μ$ of $μ$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre $x \in \mathbb{R}^{d} \setminus \operatorname{spt} μ$, outside an exceptional set of dimension at most $2(d - 1) - s$. This is sharp. In fact, for $x$ outside an exceptional set as above, $π_{x\sharp}μ\in L^{p}(S^{d - 1})$ for some $p > 1$.

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arXiv2017

Every locally finite Borel measure on $\mathbb{R}$ has conformal dimension zero

A result of P. Tukia from 1989 says that Lebesgue measure on $\mathbb{R}$ has conformal dimension zero: for every $ε> 0$, there is a Borel set $G \subset \mathbb{R}$ of full Lebesgue measure, and a quasisymmetric homeomorphism $f \colon \mathbb{R} \to \mathbb{R}$ such that $\dim_{\mathrm{H}} f(G) < ε$. In this short note, I show that the same is true for every locally finite Borel measure on $\mathbb{R}$.

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arXiv2016

Improving Kaufman's exceptional set estimate for packing dimension

Given $0 < s < 1$, I prove that there exists a constant $ε= ε(s) > 0$ such that the following holds. Let $K \subset \mathbb{R}^{2}$ be a Borel set with $\mathcal{H}^{1}(K) > 0$, and let $E_{s}(K) \subset S^{1}$ be the collection of unit vectors $e$ such that $$\dim_{\mathrm{p}} π_{e}(K) \leq s.$$ Then $\dim_{\mathrm{H}} E_{s}(K) \leq s - ε$.

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arXiv2016

The Assouad dimensions of projections of planar sets

We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension $s \in [0,2]$, then the projections have Assouad dimension at least $\min\{1,s\}$ almost surely. Compared to the famous analogue for Hausdorff dimension -- namely \emph{Marstrand's Projection Theorem} -- a striking difference is that the words `at least' cannot be dispensed with: in fact, for many planar self-similar sets of dimension $s < 1$, we prove that the Assouad dimension of projections can attain both values $s$ and $1$ for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the `discrete rotations' case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension, in which case the Assouad dimension is equal to 1. In the `dense rotations' case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no \emph{Falconer's Theorem} for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.

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arXiv2016

Curve packing and modulus estimates

A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $\mathbb{R}^{2}$ of length one. The classical "worm problem" of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family has always area at least $c$ for some small absolute constant $c > 0$. We strengthen Marstrand's result by showing that for $p > 3$, the $p$-modulus of a Moser family of curves is at least $c_{p} > 0$.

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arXiv2016

Quasisymmetric maps on Kakeya sets

I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of a.e. line segment inside $K$ has dimension at most $\min\{2n/(n + 2),7n/(4n + 3)\}$. The Kakeya maximal function conjecture implies that the bounds can be improved to $n$ and $1$, respectively.

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arXiv2016

Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $μ$, the density weighted Jones' square function $$ J_{1}(x) = \mathop{\sum_{Q \in \mathcal{D}}}_{\ell(Q) \leq 1} β_{2,μ}^{2}(3Q)\frac{\ell(Q)}{μ(Q)} 1_{Q}(x) $$ is finite for $μ$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $ε> 0$, we construct a Radon probability measure on $[0,1]^{2} \subset \mathbb{R}^{2}$ with the properties that $J_{1}(x) \leq ε$ for all $x \in \operatorname{spt} μ$, but nevertheless the $1$-dimensional lower density of $μ$ vanishes almost everywhere. In particular, $μ$ is purely $1$-unrectifiable.

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arXiv2016

Projections of planar sets in well-separated directions

First, let $K \subset B(0,1) \subset \mathbb{R}^{2}$ be a set with $\mathcal{H}_{\infty}^{1}(K) \sim 1$, and write $π_{e}(K)$ for the orthogonal projection of $K$ into the line spanned by $e \in S^{1}$. For $1/2 \leq s < 1$, write $$E_{s} := \{e : N(π_{e}(K),δ) \leq δ^{-s}\}, $$ where $N(A,r)$ is the $r$-covering number of the set $A$. It is well-known -- and essentially due to R. Kaufman -- that $N(E_{s},δ) \lessapprox δ^{-s}$. Using the polynomial method, I prove that $$ N(E_{s},r) \lessapprox \min\left\{δ^{-s}\left(\fracδ{r}\right)^{1/2},r^{-1}\right\}, \quad δ\leq r \leq 1.$$ I construct examples showing that the exponents in the bound are sharp for $δ\leq r \leq δ^{s}$. The second theorem concerns projections of $1$-Ahlfors-David regular sets. Let $A \geq 1$ and $1/2 \leq s < 1$ be given. I prove that, for $p = p(A,s) \in \mathbb{N}$ large enough, the finite set of unit vectors $S_{p} := \{e^{2πi k/p} : 0 \leq k < p\}$ has the following property. If $K \subset B(0,1)$ is non-empty and $1$-Ahlfors-David regular with regularity constant at most $A$, then $$\frac{1}{p} \sum_{e \in S_{p}} N(π_{e}(K),δ) \geq δ^{-s}$$ for all small enough $δ> 0$. In particular, $\overline{\dim}_{\text{B}} π_{e}(K) \geq s$ for some $e \in S_{p}$.

math.CA↗

arXiv2016

Characterising the big pieces of Lipschitz graphs property using projections

We characterise the big pieces of Lipschitz graphs property in terms of projections. Roughly speaking, we prove that if a large subset of an $n$-Ahlfors-David regular set $E \subset \mathbb{R}^d$ has plenty of projections in $L^{2}$, then a large part of $E$ is contained in a single Lipschitz graph. This is closely related to a question of G. David and S. Semmes.

math.CA↗

arXiv2016

A sharp exceptional set estimate for visibility

A Borel set $B \subset \mathbb{R}^{n}$ is visible from $x \in \mathbb{R}^{n}$, if the radial projection of $B$ with base point $x$ has positive $\mathcal{H}^{n - 1}$ measure. I prove that if $\dim B > n - 1$, then $B$ is visible from every point $x \in \mathbb{R}^{n} \setminus E$, where $E$ is an exceptional set with dimension $\dim E \leq 2(n - 1) - \dim B$. This is the sharp bound for all $n \geq 2$. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for $\dim E$ was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.

math.CA↗

arXiv2015

On the distance sets of AD-regular sets

I prove that if $\emptyset \neq K \subset \mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \geq 1$, then $$\dim_{\mathrm{p}} D(K) = 1,$$ where $D(K) := \{|x - y| : x,y \in K\}$ is the distance set of $K$, and $\dim_{\mathrm{p}}$ stands for packing dimension. The same proof strategy applies to other problems of similar nature. For instance, one can show that if $\emptyset \neq K \subset \mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \geq 1$, then there exists a point $x_{0} \in K$ such that $\dim_{\mathrm{p}} K \cdot (K - x_{0}) = 1$. Specialising to product sets, one derives the following sum-product corollary: if $A \subset \mathbb{R}$ is a non-empty compact $s$-Ahlfors-David regular set with $s \geq 1/2$, then $$\dim_{\mathrm{p}} [A(A - a_{1}) + A(A - a_{1})] = 1$$ for some $a_{1},a_{2} \in A$. In particular, $\dim_{\mathrm{p}} [AA + AA - AA - AA] = 1$. In all of the results mentioned above, compactness can be relaxed to boundedness and $\mathcal{H}^{s}$-measurability, if packing dimension is replaced by upper box dimension.

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