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Tuomas Orponen

Publications and source records attributed to Tuomas Orponen.

At least 91 records · Page 5Linked to original sources
arXiv2015

Hausdorff dimension, intersection of projections and exceptional plane sections

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems. Our first main result considers the orthogonal projections of two Borel sets $A,B \subset \mathbb{R}^{2}$ into one-dimensional subspaces. Under the assumptions $\dim A \leq 1 < \dim B$ and $\dim A + \dim B > 2$, we prove that the intersection of the projections $P_{L}(A)$ and $P_{L}(B)$ has dimension at least $\dim A - ε$ for positively many lines $L$, and for any $ε> 0$. This is quite sharp: given $s,t \in [0,2]$ with $s + t = 2$, we construct compact sets $A,B \subset \mathbb{R}^{2}$ with $\dim A = s$ and $\dim B = t$ such that almost all intersections $P_{L}(A) \cap P_{L}(B)$ are empty. In case both $\dim A > 1$ and $\dim B > 1$, we prove that the intersections $P_{L}(A) \cap P_{L}(B)$ have positive length for positively many $L$. If $A \subset \mathbb{R}^{2}$ is a Borel set with $0 < \mathcal{H}^{s}(A) < \infty$ for some $s > 1$, it is known that $A$ is 'visible' from almost all points $x \in \mathbb{R}^{2}$ in the sense that $A$ intersects a positive fraction of all lines passing through $x$. In fact, a result of Marstrand says that such non-empty intersections typically have dimension $s - 1$. Our second main result strengthens this by showing that the set of exceptional points $x \in \mathbb{R}^{2}$, for which Marstrand's assertion fails, has Hausdorff dimension at most one.

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arXiv2015

Hausdorff dimension estimates for restricted families of projections in $\mathbb{R}^3$

This paper is concerned with restricted families of projections in $\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently well-curved one-dimensional family of two-dimensional subspaces, the main result states that there exists $σ(s) > 1$ such that $\dim π_{V}(K) \geq σ(s)$ for almost all $V \in \mathcal{G}$. A similar result is obtained for some specific families of one-dimensional subspaces.

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arXiv2015

On the packing measure of slices of self-similar sets

Let $K \subset \mathbb{R}^{2}$ be a rotation and reflection free self-similar set satisfying the strong separation condition, with dimension $\dim K = s > 1$. Intersecting $K$ with translates of a fixed line, one can study the $(s - 1)$-dimensional Hausdorff and packing measures of the generic non-empty line sections. In a recent article, T. Kempton gave a necessary and sufficient condition for the Hausdorff measures of the sections to be positive. In this paper, I consider the packing measures: it turns out that the generic section has infinite $(s - 1)$-dimensional packing measure under relatively mild assumptions.

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arXiv2014

On the tube-occupancy of sets in $\mathbb{R}^d$

Call a pair $(s,t) \in [0,d] \times [0,d]$ admissible, if there exists a compact set $K \subset \mathbb{R}^{d}$ and a constant $C > 0$ such that $0 < \mathcal{H}^{s}(K) < \infty$, and $$\mathcal{H}_{s}(K \cap T) \leq Cw(T)^{t}$$ for all tubes $T \subset \mathbb{R}^{d}$ of width $w(T)$. The purpose of this paper is to show that all pairs $(s,s)$ with $s < 1$ are admissible. Combined with previous results, this settles a question of A. Carbery.

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arXiv2014

A discretised projection theorem in the plane

The main result of this paper is that for any $1/2 \leq s < 2 - \sqrt{2} \approx 0.5858$, there is a number $σ= σ(s) < s$ with the following property. Let $δ> 0$ be small, assume that $A \subset [0,1]$ is a $(δ,1/2)$-set, and that $E \subset [0,1]$ contains $\gtrsim δ^{-σ}$ roughly $δ^{s}$-separated points. Then there exists a number $t \in E$ such that $A + tA$ contains $\gtrsim δ^{-s}$ $δ$-separated points. For $σ= s$, this is essentially a consequence of Kaufman's well-known bound for exceptional sets of projections. Our proof consists of a structural observation concerning sets, for which Kaufman's bound is near-optimal, combined with (an adaptation of) Solymosi's argument for his "$4/3$" sum-product theorem.

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arXiv2014

Some obstacles in characterising the boundedness of bi-parameter singular integrals

The famous $T1$ theorem for classical Calderón-Zygmund operators is a characterisation for their boundedness in $L^{2}$. In the bi-parameter case, on the other hand, the current $T1$ theorem is merely a collection of sufficient conditions. This difference in mind, we study a particular dyadic bi-parameter singular integral operator, namely the full mixed bi-parameter paraproduct $P$, which is precisely the operator responsible for the outstanding problems in the bi-parameter theory. We make several remarks about $P$, the common theme of which is to demonstrate the delicacy of the problem of finding a completely satisfactory product $T1$ theorem. For example, $P$ need not be unconditionally bounded if it is conditionally bounded -- a major difference compared to the corresponding one-parameter model operators. Moreover, currently the theory even lacks a characterisation for the potentially easier unconditional boundedness. The product BMO condition is sufficient, but far from necessary: we show by example that unconditional boundedness does not even imply the weaker rectangular BMO condition.

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arXiv2014

On Ahlfors-David regular weighted bounds for the extension operator associated to the circle

This paper addresses the sharpness of a weighted $L^{2}$-estimate for the Fourier extension operator associated to the circle, obtained by J. Bennett, A. Carbery, F. Soria and A. Vargas in 2006. A point left open in their paper was the necessity of a certain $\log R$-factor in the bound. Here, I show that the factor is necessary for all $1/2$-Ahlfors-David regular weights on the circle, but it can be removed for $s$-Ahlfors-David regular weights with $s \neq 1/2$.

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arXiv2014

On restricted families of projections in R^3

We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most $1/2$-dimensional sets $B \subset \mathbb{R}^{3}$ is typically preserved under one-dimensional families of projections onto lines. We improve the result by an $\varepsilon$, proving that if $\dim_{\mathrm{H}} B = s > 1/2$, then the packing dimension of the projections is almost surely at least $σ(s) > 1/2$. For projections onto planes, we obtain a similar bound, with the threshold $1/2$ replaced by $1$. In the special case of self-similar sets $K \subset \mathbb{R}^{3}$ without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The $\dim_{\mathrm{H}} K \leq 1$ case of the result follows from recent work of M. Hochman, but the $\dim_{\mathrm{H}} K > 1$ part is new: with this assumption, we prove that the projections have positive length almost surely.

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arXiv2013

On the packing measure of self-similar sets

Building on a recent result of M. Hochman, we give an example of a linear self-similar set K such that 0 < dim K = s < 1 and P^s(K) = 0. This answers a question of Y. Peres and B. Solomyak.

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arXiv2013

Square functions with general measures II

We continue developing the theory of conical and vertical square functions on $R^{n}$, where $μ$ is a power bounded measure, possibly non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove a general local $Tb$ theorem with tent space $T^{2,\infty}$ type testing conditions to characterise the $L^{2}$ boundedness. Second, we completely answer the question, whether the boundedness of our operators on $L^{2}$ implies boundedness on other $L^{p}$ spaces, including the endpoints. For the conical square function, the answers are generally affirmative, but the vertical square function can be unbounded on $L^{p}$ for $p > 2$, even if $μ= dx$. For this, we present a counterexample. Our kernels $s_t$, $t > 0$, do not necessarily satisfy any continuity in the first variable -- a point of technical importance throughout the paper. Third, we construct a non-doubling Cantor-type measure and an associated conical square function operator, whose $L^{2}$ boundedness depends on the exact aperture of the cone used in the definition. Thus, in the non-homogeneous world, the 'change of aperture' technique -- widely used in classical tent space literature -- is not available. Fourth, we establish the sharp $A_{p}$-weighted bound for the conical square function under the assumption that $μ$ is doubling.

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arXiv2013

Constancy results for special families of projections

Let {\mathbb{V} = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let π_{\mathbb{V}} : R^n --> \mathbb{V} be the orthogonal projection onto \mathbb{V}. We prove that the mapping V \mapsto Dim π_{\mathbb{V}}(B) is almost surely constant for any analytic set B \subset R^n, where Dim denotes either Hausdorff or packing dimension.

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arXiv2013

On the Haar Shift representations of Calderón-Zygmund Operators

In connection with proving the A_2 conjecture in 2010, T. Hytönen obtained a representation of general Caldéron-Zygmund operators in terms of simpler operators known as Haar shifts. In this note, we prove that the result is sharp in the sense that Haar shift representations of Hytönen's type are only available for Calderón-Zygmund operators.

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arXiv2013

On Fourier analytic properties of graphs

We study the Fourier dimensions of graphs of real-valued functions defined on the unit interval [0,1]. Our results imply that the graph of the fractional Brownian motion is almost surely not a Salem set, answering in part a question of Kahane from 1993, and that the graph of a Baire typical function in C[0,1] has Fourier dimension zero.

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arXiv2012

Slicing Sets and Measures, and the Dimension of Exceptional Parameters

We consider the problem of slicing a compact metric space Ωwith sets of the form π_λ^{-1}\{t\}, where the mappings π_λ \colon Ω\to \R, λ\in \R, are \emph{generalized projections}, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that Ωhas Hausdorff dimension strictly greater than one, what is the dimension of the 'typical' slice π_λ^{-1}{t}, as the parameters λand t vary. In the special case of the mappings π_λ being orthogonal projections restricted to a compact set Ω\subset \R^{2}, the problem dates back to a 1954 paper by Marstrand: he proved that for almost every λthere exist positively many $t \in \R$ such that \dim π_λ^{-1}{t} = \dim Ω- 1. For generalized projections, the same result was obtained 50 years later by Järvenpää, Järvenpää and Niemelä. In this paper, we improve the previously existing estimates by replacing the phrase 'almost all λ' with a sharp bound for the dimension of the exceptional parameters.

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