arXiv subjects
Publications and source records attributed to Pertti Mattila.
For Borel sets $A\subset \mathbb{R}^n\times \mathbb{R}$ we prove lower bounds for the parabolic Hausdorff dimension of the orthogonal projections of $A$ on generic $m$-dimensional linear subspaces of $\mathbb{R}^n\times \mathbb{R}$.
For Borel subsets $Θ\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*} Θ(E):=\bigcup_{(g,z)\in Θ}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and Hausdorff dimension of $Θ(E)$ given the dimensions of the Borel sets $E$ and $Θ$, when $Θ$ has product form. We also study this question by replacing rigid motions with the class of dilations and translations; and similarity transformations. The dimensional thresholds are sharp. Our results are variants of some previously known results in the literature when $E$ is restricted to smooth objects such as spheres, $k$-planes, and surfaces.
This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_λ:\mathbb{R}^{n}\to\mathbb{R}^{m}, λ\in Λ,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_λ^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset\mathbb{R}^{n}$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$. As an application we prove for measurable sets $A,B\subset\mathbb{R}^{n}$ with positive $s$- and $t$-dimensional measures that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) \geq s+t - n$ for almost all rotations $g$ and for positively many $z\in\mathbb{R}^{n}$. We shall also give an application on the estimates of the dimension of the set of exceptional rotations.
Let $A$ and $B$ be Borel subsets of the Euclidean $n$-space with $\dim A + \dim B > n$. This is a survey on the question: what can we say about the Hausdorff dimension of the intersections $A\cap (g(B)+z)$ for generic orthogonal transformations $g$ and translations by $z$.
This is a survey on rectifiability. I discuss basic properties of rectifiable sets, measures, currents and varifolds and their role in complex and harmonic analysis, potential theory, calculus of variations, PDEs and some other topics.
We define rectifiability in $\mathbb{R}^{n}\times\mathbb{R}$ with a parabolic metric in terms of $C^1$ graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent measures. We also discuss relations between the parabolic rectifiability and other notions of rectifiability.
For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of $\mathbb{R}^{2n}$ and for product sets.
We give conditions on a general family $P_λ:\R^n\to\R^m, λ\in Λ,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_λ^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset\Rn$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,B\subset\Rn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $z\in\Rn$.
This is a survey on recent developments on the Hausdorff dimension of projections and intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the Hausdorff dimension of exceptional sets and on restricted projection families. We shall also discuss relations between projections and Hausdorff dimension of Besicovitch sets.
We prove some geometric properties of sets in the first Heisenberg group whose Heisenberg Hausdorff dimension is the minimal or maximal possible in relation to their Euclidean one and the corresponding Hausdorff measures are positive and finite. In the first case we show that these sets must be in a sense horizontal and in the second case vertical. We show the sharpness of our results with some examples.
For Borel subsets A and B of the Euclidean n-space the intersection of A with generic rotations and translations of B has often Hausdorff dimension at least dim A + dim B - n. Estimates for the exceptional set of rotations are derived.
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 that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.
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.
We investigate certain singular integral operators with Riesz-type kernels on s-dimensional Ahlfors-David regular subsets of Heisenberg groups. We show that $L^2$-boundedness, and even a little less, implies that $s$ must be an integer and the set can be approximated at some arbitrary small scales by homogeneous subgroups. It follows that the operators cannot be bounded on many self similar fractal subsets of Heisenberg groups.
In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.
We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.
We shall consider the truncated singular integral operators T_{μ, K}^εf(x)=\int_{\mathbb{R}^{n}\setminus B(x,ε)}K(x-y)f(y)dμy and related maximal operators $T_{μ,K}^{\ast}f(x)=\underset{ε>0}{\sup}| T_{μ,K}^εf(x)|$. We shall prove for a large class of kernels $K$ and measures $μ$ and $ν$ that if $μ$ and $ν$ are separated by a Lipschitz graph, then $T_{ν,K}^{\ast}:L^p(ν)\to L^p(μ)$ is bounded for $1<p<\infty$. We shall also show that the truncated operators $T_{μ, K}^ε$ converge weakly in some dense subspaces of $L^2(μ)$ under mild assumptions for the measures and the kernels.
We give sufficient conditions to guarantee that if two self-conformal sets E and F have Lipschitz equivalent subsets of positive measure, then there is a bilipschitz map of E into, or onto, F.