Related papers: On Borel hull operations
In important papers on cake-cutting -- one of the key areas in fair division and resource allocation -- the measure-theoretical fundamentals are not fully correctly given. It is not clear (i) which family of sets should be taken for the…
A real number $x$ is considered normal in an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', ensuring that for each $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the…
The classical Brauer-Siegel conjecture describes the asymptotic behaviour of the product of the class number and the regulator in families of number fields. All known cases of the conjecture rely on reducing the problem, via group theoretic…
In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the…
We prove that the relation of bisimilarity between countable labelled transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on…
Motivated by recent investigations of Sophie Grivaux and \'Etienne Matheron on the existence of invariant measures in Linear Dynamics, we introduce the concept of locally bounded orbit for a continuous linear operator $T:X\longrightarrow X$…
We prove that an homogeneous space of an almost simple Lie group does not have any measurable tiling invariant by a lattice of the Lie group. This refines the Howe-Moore ergodicity theorem. -- On d\'emontre qu'un espace homog\`ene d'un…
One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space…
Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \supset X$ and a Borel probability measure $\mu$ on $G$ such that…
The article has been withdrawn by the author. Wolfgang Lueck and Peter Linnell pointed out that the proof of Lemma 3.8 does not apply to the unrestricted case of wreath product. It is not clear at this stage how to complete the proof of…
We study and classify topologically invariant $\sigma$-ideals with a Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal…
We present a general method of constructing an uncountable family of regular Borel measures on certain path spaces of Lipschitz functions having fixed Lipschitz constants. We use this method to give a definition of Lebesgue measure and…
We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of…
Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \sigma-finite measure to $B$. We…
We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota-Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this…
The Steinhaus-Weil theorem that concerns us here is the simple, or classical, `interior-points' property -- that in a Polish topological group a non-negligible set B has the identity as an interior point of $BB^{-1}$. There are various…
Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel…
We obtain, under an additional assumption on the subanalytic abnormal distribution constructed in [4], a proof of the minimal rank Sard conjecture in the analytic category. It establishes that from a given point the set of points accessible…
We propose a reformulation of the ideal $\mathcal{N}$ of Lebesgue measure zero sets of reals modulo an ideal $J$ on $\omega$, which we denote by $\mathcal{N}_J$. In the same way, we reformulate the ideal $\mathcal{E}$ generated by…
Standard results in descriptive set theory provide sufficient conditions for a Borel set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ to admit a Borel uniformization, namely, when $P$ has "small" sections or "large"…