Related papers: Comparing the uniformity invariants of null sets f…
This note describes Fatou's lemma and Lebesgue's dominated convergence theorem for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to…
It is well-known that for every $N \geq 1$ and $d \geq 1$ there exist point sets $x_1, \dots, x_N \in [0,1]^d$ whose discrepancy with respect to the Lebesgue measure is of order at most $(\log N)^{d-1} N^{-1}$. In a more general setting,…
The paper, that continuous some previous work of Sch\"onherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive…
In this work, we study a numeral system with a natural base $s \geq 2$ and a redundant alphabet $A_r=\{0,1, \dots, r\}$, where $s \leq r \leq 2s-2$. We investigate the topological, metric, and fractal properties of the set of numbers in the…
We show that the uniform Littlewood Conjecture (ULC) recently introduced by Bandi, Fregoli and Kleinbock is false. More precisely the counterexamples form a residual set, the method further suggests positive Hausdorff dimension. For a…
Falconer proved that there are sets $E\subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ whose distance sets $\{|x-y| : x,y\in E\}$ are null with respect to Lebesgue measure. This led to the conjecture that distance sets have positive…
Continuity of measure asserts that the measure of the union of an increasing sequence of sets is equal to the supremum of the measures of those sets. We provide counter examples in the case of uncountable unions. We construct the first…
The smooth development of large parts of mathematics hinges on the idea that some sets are `small' or `negligible' and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely `measure zero', originated…
In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…
A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic…
We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and…
We study the consistency strength of Lebesgue measurability for $\Sigma^1_3$ sets over Zermelo set theory ($Z$) in a completely choiceless context. We establish a result analogous to the Solovay-Shelah theorem.
We investigate the measurement incompatibility of continuous-variable systems with infinite-dimensional Hilbert spaces under the influence of pure losses, a fundamental noise source in quantum optics, and a significant challenge for…
This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge…
We extend some of the measures of association defined by Lazarsfeld and Martin, obtaining useful invariants to compare the birational geometry of two varieties having different dimensions. We explore such invariants providing examples and…
We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of…
In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove \begin{align*} \|f(A)Xg(B)\pm…
We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval $T_{\beta}(x)=\beta x$ (mod $1$), $x\in[0,1]$, $\beta>0$, which are the so-called…
We prove that every hyperbolic measure invariant under a C^{1+\alpha} diffeomorphism of a smooth Riemannian manifold possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the…
We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\rho$ become 'smaller' as $\rho\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function…