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We answer a question of Banakh, Jabłońska and Jabłoński by showing that for d ⩾ 2 there exists a compact set K ⊆ R d such that the projection of K onto each hyperplane is of non-empty interior, but K + K is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a K in the unit cube with full projections, that is, such that the projections of K agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for ℓ -fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar. [ABSTRACT FROM AUTHOR]

The lower dimension \dim _L is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu [Adv. Math. 329 (2018), pp. 273–328] introduced the modified lower dimension dim_\textit {{ML}} by making the lower dimension monotonic with the simple formula dim_\textit {{ML}}X=\sup \{\dim _L E: E\subset X\}. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space X let \mathcal {K}(X) denote the metric space of the non-empty compact subsets of X endowed with the Hausdorff metric. As an application of our characterization, we show that the map dim_\textit {{ML}}\colon \mathcal {K}(X)\to [0,\infty ] is Borel measurable. More precisely, it is of Baire class 2, but in general not of Baire class 1. This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of \ell ^1 endowed with the Effros Borel structure. [ABSTRACT FROM AUTHOR]

A subset X of a Polish group G is 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 ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky. [ABSTRACT FROM AUTHOR]

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 were already known for Haar null sets. [ABSTRACT FROM AUTHOR]

In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood; see, e.g., the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For Homeo+([0, 1]) we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for Homeo+(S1) we describe the non-Haar null conjugacy classes, and for U(l2) we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set. [ABSTRACT FROM AUTHOR]

In order to understand the structure of the “typical” element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski, we investigate the measure theoretic dual of this problem, using Christensen’s notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper we generalize the theorems of Dougherty and Mycielski about S∞ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the cofinal strong amalgamation property. As an application, we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set. [ABSTRACT FROM AUTHOR]

We show that the σ-ideal of Borel subsets of R³ of σ-finite 2-dimensional Hausdorff measure is not homogeneous. This partially answers a question of Zapletal. We prove that each of the statements cov(N) < cov(N¹2), cov(N¹2) < non(M), and cov(M) < non(N¹2) is consistent, where N¹ 2 is the σ-ideal of sets in the plane of 1-dimensional Hausdorff measure zero, and N and M are the usual null and meagre σ-ideals. This answers a question of Fremlin and settles the question of strictness of all the inequalities once we fit the cardinal invariants of N¹2 into the Cichón diagram. We prove that it is consistent that there is an ordering of the reals in which all proper initial segments are Lebesgue null, but for every ordering of the reals there is a proper initial segment that is not null with respect to the 1/2-dimensional Hausdorff measure. This answers a question of Humke and Laczkovich. [ABSTRACT FROM AUTHOR]

A set is shy or Haar null (in the sense of Christensen) if there exists a Borel set and a Borel probability measure μ on C[0, 1] such that and for all f ∈ C[0, 1]. The complement of a shy set is called a prevalent set. We say that a set is Haar ambivalent if it is neither shy nor prevalent.The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy many f ∈ C[0, 1]?The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category) f ∈ C[0, 1] from the topological point of view. We prove that the functions f ∈ C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper, Balka et al. proved that the functions f ∈ C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set in C[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions f ∈ C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦ f–1 are not perfect form a Haar ambivalent set in C[0, 1].We show that for the prevalent f ∈ C[0, 1] for the generic y ∈ f([0, 1]) the level set f–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functions f ∈ C[0, 1] for which there exists a perfect set Pf ⊂ [0, 1] such that f'(x) = ∞ for all x ∈ Pf is Haar ambivalent. [ABSTRACT FROM PUBLISHER]

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in [ABSTRACT FROM AUTHOR]

Let G be an abelian Polish group, e.g., a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure µ on G such that µ( B + g) = 0 for every g ∈ 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 $${G_\delta }$$ Haar null set. We also show that $${G_\delta }$$ can be replaced by any other class of the Borel hierarchy, which implies that the additivity of the σ-ideal of Haar null sets is ω. The definition of a generalised Haar null set is obtained by replacing the Borelness of B in the above definition by universal measurability. We give an example of a generalised Haar null set that is not Haar null, more precisely, we construct a coanalytic generalised Haar null set without a Borel Haar null hull. This solves Problem GP from Fremlin's problem list. Actually, all our results readily generalise to all Polish groups that admit a two-sided invariant metric. [ABSTRACT FROM AUTHOR]

The main goal of this note is to prove the following theorem. If An is a sequence of measurable sets in a δ-finite measure space (X, A, μ) that covers μ-a.e. x ∈ X infinitely many times, then there exists a sequence of integers ni of density zero so that Ani still covers μ-a.e. x ∈ X infinitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co-infinite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas. [ABSTRACT FROM AUTHOR]

Let K ⊂ ℝd be a self-similar or self-affine set and let μ be a self-similar or self-affine measure on it. Let G be the group of affine maps, similitudes, isometries or translations of ℝd . Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or that K is a so-called Sierpínski sponge) we prove theorems of the following types, which are closely related to each other. • (Non-stability) There exists a constant c < 1 such that for every g ϵ G we have either μ.K ∩ g (K)) < c · μ(K) or K ⊂ g(K). • (Measure and topology) For every g ϵ G we have μ(K) g(K)) > 0 ⇔ intK (K ∩ g(K)) ≠θ ; (where intK is interior relative to K). • (Extension) The measure μ has a G-invariant extension to ℝd . Moreover, in many situations we characterize those g for which μ;(K ∩ g(K)) > θ. We also obtain results about those g for which g(K) ⊂ K or g(K) ⊃ K. [ABSTRACT FROM AUTHOR]

A function ƒ : ℝn → ℝ is called vertically rigid if graph(cf) is isometric to graph(ƒ) for all c ≠ 0. In [1] we settled Jankovic's conjecture by showing that a continuous function ƒ : ℝ → ℝ is vertically rigid if and only if it is of the form a + bx or a + bekx (a; b; k ∈ ℝ). Now we prove that a continuous function ƒ : ℝ² → ℝ is vertically rigid if and only if, after a suitable rotation around the z-axis, ƒ (x; y) is of the form a+bx+dy, a+s(y)ekx or a+bekx +dy (a; b; d; k ∈ ℝ, k ≠ 0, s : ℝ → ℝ continuous). The problem remains open in higher dimensions. [ABSTRACT FROM AUTHOR]

A function f : ℝ → ℝ is called vertically rigid, if graph(c f) is isometric to graph(f) for all c ≠ 0. We confirm Jankovid's conjecture by showing that a continuous function is vertically rigid if and only if it is of the form a + bx or a + bekx (a,b,k ∈ ℝ). We answer a question of Cain, Clark and Rose in the negative by showing that there exists a Borel measurable vertically rigid function which is not of the above form. We discuss the Lebesgue and Baire measurable case. We also introduce horizontally rigid functions, and show that a certain structure theorem can be proved without assuming any regularity. [ABSTRACT FROM AUTHOR]

Motivated by a problem in discrete geometry we investigate the following set of problems. Let X be a set, κ a cardinal number, and H a family that covers each x ∈ X at least κ times. Under what assumptions can we decompose H into k many subcovers? Equivalently, under what assumptions can we colour H by κ many colours so that for each x ∈ X and each colour c there exists H ∈ H of colour c containing x? The assumptions we make can be e.g. that H consists of open, closed, compact, Gσ or convex subsets of ℝn, or intervals in a linearly ordered set, or we can make various restrictions on the cardinality of X, H or elements of H. Besides numerous positive and negative results many questions turn out to be independent of the axioms of set theory. [ABSTRACT FROM AUTHOR]

We show that the Continuum Hypothesis implies that for every 0 < d1 ≤ d2 < n the measure spaces (ℝn, MHd1, Hd1 and (ℝn, MHd2, Hd2) are isomorphic, where Hd is d-dimensional Hausdorff measure and MHd is the σ-algebra of measurable sets with respect to Hd. This is motivated by the well-known question (circulated by D. Preiss) whether such an isomorphism exists if we replace measurable sets by Borel sets. We also investigate the related question whether every continuous function (or the typical continuous function) is Hölder continuous (or is of bounded variation) on a set of positive Hausdorff dimension. [ABSTRACT FROM AUTHOR]