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Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text]. 

A long-standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag–Solitar group BS(1, 2) and the lamplighter group Z2 ≀ Z. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov’s notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Følner tilings, topological dynamics, and C∗-algebras. 

Abstract Schmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $$\mathsf {AD}_{\mathbb R}$$ , which is a much stronger axiom than that asserting all integer games are determined, $$\mathsf {AD}$$ . One of our main results is a general theorem which under the hypothesis $$\mathsf {AD}$$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $$(\alpha ,\beta ,\rho )$$ game on $$\mathbb R$$ is determined from $$\mathsf {AD}$$ alone, but on $$\mathbb R^n$$ for $$n \geq 3$$ we show that $$\mathsf {AD}$$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $$(\alpha , \beta , \rho )$$ game on $$\mathbb {R}$$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $$\mathsf {AD}$$ . 

Abstract A Cantor series expansion for a real number x with respect to a basic sequence $$Q=(q_1,q_2,\dots )$$ , where $$q_i \geq 2$$ , is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a $$\boldsymbol {\Pi }^0_3$$ -complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is $$\boldsymbol {\Pi }^0_3$$ -complete, and if Q is $$1$$ -divergent then the sets of normal and ratio normal numbers are $$\boldsymbol {\Pi }^0_3$$ -complete. We further show that all five non-trivial differences of these sets are $$D_2(\boldsymbol {\Pi }^0_3)$$ -complete if $$\lim _i q_i=\infty $$ and Q is $$1$$ -divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible. 

Abstract We consider the complexity of special $$\alpha $$ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha. 

We introduce the notion of [Formula: see text]-determinacy for [Formula: see text] a pointclass and [Formula: see text] an equivalence relation on a Polish space [Formula: see text]. A case of particular interest is the case when [Formula: see text] is the (left) shift-action of [Formula: see text] on [Formula: see text] where [Formula: see text] or [Formula: see text]. We show that for all shift actions by countable groups [Formula: see text], and any “reasonable” pointclass [Formula: see text], that [Formula: see text]-determinacy implies [Formula: see text]-determinacy. We also prove a corresponding result when [Formula: see text] is a subshift of finite type of the shift map on [Formula: see text]. 

Abstract Let $$\mathcal {N}(b)$$ be the set of real numbers that are normal to base b . A well-known result of Ki and Linton [19] is that $$\mathcal {N}(b)$$ is $$\boldsymbol {\Pi }^0_3$$ -complete. We show that the set $${\mathcal {N}}^\perp (b)$$ of reals, which preserve $$\mathcal {N}(b)$$ under addition, is also $$\boldsymbol {\Pi }^0_3$$ -complete. We use the characterization of $${\mathcal {N}}^\perp (b),$$ given by Rauzy, in terms of an entropy-like quantity called the noise . It follows from our results that no further characterization theorems could result in a still better bound on the complexity of $${\mathcal {N}}^\perp (b)$$ . We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $$\boldsymbol {\Pi }^0_4$$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.