Márton Elekes, Márk Poór, Zoltán Vidnyánszky
We show that every non-Haar-null analytic subset of $\mathbb{Z}^ω$ contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of $\mathbb{Z}^ω$ form a $\mathbfΔ^1_2$ set.
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We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which …
Following Darji, we say that a Borel subset $B$ of an abelian Polish group $G$ is Haar meager if there is a compact metric space $K$ and a continuous function $f : K \to G$ such that the preimage of the translate, $f^{-1}(B+g)$ is meager in $K$ for every $g \in G$. The set $B$ is called strongly Haar meager if there is a compact set $C \subseteq G$ such that $(B+g) \cap C$ is meager in $C$ for every $g \in G$. The main open problem in this area is Darji's question asking whether these two notions are the same. Even though there have been several partial results suggesting a positive answer, in…
A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $μ$ on $G$ such that $μ(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $μ$ such that $μ(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This resul…
We show that for every $2\le ξ<ω_1$ there exists a Haar null set in $\mathbb{Z}^ω$ that is the difference of two $\mathbfΠ^0_ξ$ sets but not contained in any $\mathbfΠ^0_ξ$ Haar null set. In particular, there exists a Haar null set in $\mathbb{Z}^ω$ that is the difference of two $G_δ$ sets but not contained in any $G_δ$ Haar null set. This partially answers a question of M. Elekes and Z. Vidnyánszky. To prove this, we also prove a theorem which characterizes the Haar null subsets of $\mathbb{Z}^ω$.
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 $μ$ on $G$ such that $μ(B+g)=0$ for every $g \in G$. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_δ$ Haar null set. We also show that $G_δ$ can be replaced by any other class of…
Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gXh)=0$. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.
A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $μ$ and a Borel set $B$ containing $X$ such that $μ(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $σ$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}…
Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure 0. Under an extra technical hypothesis, this also holds if X is constructible. We deduce from this generic compact domination for definably amenable NIP groups.
We study `definable' subsets of Baire space $\mathcal{N}$. The logic of our arguments is intuitionistic and we use L.E.J.~Brouwer's Thesis on bars in $\mathcal{N}$ and his continuity axioms. We avoid the operation of taking the complement of a subset of $\mathcal{N}$. A subset of $\mathcal{N}$ is $\mathbfΣ^1_1$ or: analytic if it is the projection of a closed subset of $\mathcal{N}$. Important $\mathbfΣ^1_1$ set are the set of the codes of all closed and located subsets of $\mathcal{N}$ that are positively uncountable and the set of the codes of all located and closed subsets of $\mathcal{N}$ …
Measure doubling of small sets in $\mathrm{SO}(3,\mathbb{R})$
本文解决了 Breuillard 和 Green 提出的问题，证明在三维旋转群 SO(3,R) 中，测度足够小的开集 A 满足 μ(A^2) > 3.99 μ(A)，并将该结果推广为两个集合乘积的 Brunn-Minkowski 型不等式。
On relatively analytic and Borel subsets
定义z为函数f：X-> Y的最小基数，其中X和Y是实数集，使得没有Borel函数g扩展f。在本文中，我们证明了它是相对一致的ZFC有b <z，其中b是，通常情况下，最小的基数的一个无界的家庭在w ^w。这回答了Zapletal提出的一个问题。我们还证明了存在一个实数集X使得X的Borel序是有界的，但存在X的一个相对解析子集，它不是相对余解析的，这与ZFC是相对一致的.这回答了莫尔丁的一个问题。
Subanalytic sets and complex analytic geometry
考虑复解析流形M（当M被看作实流形时）的次解析子集A，并给出了A是M的复解析子集的条件.
We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for $Σ^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\mathcal{U}$ is the set of all pairs $( K , r )$ with $K$ compact and $r \in ( 0 ; 1 )$ being the density of some point with respect to $K$. This result yields that the set of all $K$ such that the range of its density function is $S \cup \{ 0 , 1 \}$, for some fixed uncountable analytic set $S \subseteq ( 0 ; 1 )$, is $Π^{1}_{2}$-complete.
Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $σ$-additive measure $μ$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $μ(U)<μ(V)$ there is a clopen set $W\subset V$ with $μ(W)=μ(U)$. We study $σ$-ideals with Borel base on $X$ which are invariant under the action of the group $H_μ(X)$ of measure-preserving homeomorphisms of $(X,μ)$, and show that any such $σ$-ideal $\mathcal I$ is equal to one of seven $σ$-ideals: $\{\emptyset\}$, $[X]^{\leω}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal…
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