We consider classes of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim sup En ⊂ [0, 1], and that μn are probability measures with support in En. If there exists a constant C such that

for all n, then, under suitable conditions on the limit measure of the sequence (μn), we prove that the set E is in the class .

As an application we prove that, for α > 1 and almost all λ ∈ (½, 1), the set

where and ak ∈ {0, 1}}, belongs to the class . This improves one of our previously published results.

Let (E, ℱ) be a weakly compactly generated Frechet space and let ℱ0 be another weaker Hausdorff locally convex topology on E. Let X be an ℱ-bounded compact subset of (E, ℱ0). The ℱ0-closed convex hull of X in E is then ℱ0-compact. We also give a new proof, without using Riemann–Lebesgue-integrable (Birkoff-integrable) functions, with the result that if (E, ∥ · ∥) is any Banach space and ℱ0 is fragmented by ∥ · ∥, then the same result holds. Furthermore, the closure of the convex hull of X in ℱ0-topology and in the original topology of E is the same.

Let $Q$ be an infinite subset of $\mathbb{N}$. For any ${\it\tau}>2$, denote $W_{{\it\tau}}(Q)$ (respectively $W_{{\it\tau}}$) to be the set of ${\it\tau}$ well-approximable points by rationals with denominators in $Q$ (respectively in $\mathbb{N}$). We consider the Hausdorff dimension of the liminf set $W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ after Adiceam. By using the tools of continued fractions, it is shown that if $Q$ is a so-called $\mathbb{N}\setminus Q$-free set, the Hausdorff dimension of $W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ is the same as that of $W_{{\it\tau}}$, i.e. $2/{\it\tau}$.

We study the distribution of the orbits of real numbers under the beta-transformation $T_{{\it\beta}}$ for any ${\it\beta}>1$. More precisely, for any real number ${\it\beta}>1$ and a positive function ${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:

$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

An extension of Rademacher’s theorem is proved for Lipschitz mappings between Banach spaces without the Radon–Nikodým property.

We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas of Kieninger [Iterated Function Systems on Compact Hausdorff Spaces (Shaker, Aachen, 2002)] and McGehee and Wiandt [‘Conley decomposition for closed relations’, Differ. Equ. Appl. 12 (2006), 1–47] restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred invariant sets. This allows us to give short proofs of some of the key ideas.

In this paper, we study the properties of $k$-plurisubharmonic functions defined on domains in ${ \mathbb{C} }^{n} $. By the monotonicity formula, we give an alternative proof of the weak continuity of complex $k$-Hessian operators with respect to local uniform convergence.

We construct dense Borel measurable subgroups of Lie groups of intermediate Hausdorff dimension. In particular, we generalize the Erdős–Volkmann construction [Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math. 221 (1966), 203–208], showing that any nilpotent $\sigma $-compact Lie group $N$ admits dense Borel subgroups of arbitrary dimension between zero and $\dim N$. In algebraic groups defined over a finite extension of the rationals, using diophantine properties of algebraic numbers, we are also able to construct dense subgroups of arbitrary dimension, but the general case remains open. In particular, we raise the following question: does there exist a measurable proper subgroup of $ \mathbb{R} $ of positive Hausdorff dimension which is stable under multiplication by a transcendental number? Subgroups of nilpotent $p$-adic analytic groups are also discussed.

For and α, we consider sets of numbers x such that for infinitely many n, x is 2−αn-close to some ∑ ni=1ωiλi, where ωi∈{0,1}. These sets are in Falconer’s intersection classes for Hausdorff dimension s for some s such that −(1/α)(log λ /log   2 )≤s≤1/α. We show that for almost all , the upper bound of s is optimal, but for a countable infinity of values of λ the lower bound is the best possible result.

We follow the idea of generalising the notion of classical iterated function systems, as presented by Mihail and Miculescu. We give their deliberations a more general setting and, using this general approach, study the generic aspect of the problem of existence of an attractor of a function system.

The self-affine measure μM, D corresponding to M = diag[p1, p2, p3] (pj ∈ ℤ \ {0, ± 1}, j = 1, 2, 3) and D = {0, e1, e2, e3} in the space ℝ3 is supported on the three-dimensional Sierpinski gasket T(M, D), where e1, e2, e3 are the standard basis of unit column vectors in ℝ3. We shall determine the spectrality and non-spectrality of μM, D, and show that if pj ∈ 2ℤ \ {0, 2} for j = 1, 2, 3, then μM, D is a spectral measure, and if pj ∈ (2ℤ + 1) \ {±1} for j = 1, 2, 3, then μM, D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM, D), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measures.

We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are one dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.

A condition equivalent to sparseness of a set on the plane is formulated and used as a motivation for a new concept of density point on the plane. This is investigated and compared with known previous versions.

The paper extends the fundamental existence assertion for probability contents and measures with given marginals: the extension is from algebras to lattices, and thus is in accord with an actual trend in measure and integration. The proof of the basic theorem is a rapid application of a former Hahn–Banach type separation theorem.

The Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.

For a probability measure μ on a real separable Hilbert space H, we are interested in “volume-based” approximations of the d-dimensional least squares error of μ, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices in H. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of μ, where the comparison depends on a simple quantitative geometric property of μ. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of μ. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo singular value decomposition based on volume sampling.

We study a Dirichlet problem involving the weak Laplacian on the Sierpiński gasket, and we prove the existence of at least two distinct nontrivial weak solutions using Ekeland’s Variational Principle and standard tools in critical point theory combined with corresponding variational techniques.

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ1,μ2,…,μd in ℝd such that μi(ℝd)=k for all i, there is a partition of ℝd into k interior-disjoint convex parts C1,C2,…,Ck such that μi (Cj)=1 for all i,j. If k=2 , this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.

The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E) is K-analytic if and only if σ(E′,E) is Lindelöf. We prove a corresponding result for spaces Cp (X) of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X) is web-bounded. Hence the weak* dual of Cp (X) is a Lindelöf Σ-space if and only if Cp (X) is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

We discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space (). We prove that in most cases such a map is no more than 1/p-Hölder continuous.