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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 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 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. 

Abstract Wireframe DNA origami assemblies can now be programmed automatically from the top-down using simple wireframe target geometries, or meshes, in 2D and 3D, using either rigid, six-helix bundle (6HB) or more compliant, two-helix bundle (DX) edges. While these assemblies have numerous applications in nanoscale materials fabrication due to their nanoscale spatial addressability and high degree of customization, no easy-to-use graphical user interface software yet exists to deploy these algorithmic approaches within a single, standalone interface. Further, top-down sequence design of 3D DX-based objects previously enabled by DAEDALUS was limited to discrete edge lengths and uniform vertex angles, limiting the scope of objects that can be designed. Here, we introduce the open-source software package ATHENA with a graphical user interface that automatically renders single-stranded DNA scaffold routing and staple strand sequences for any target wireframe DNA origami using DX or 6HB edges, including irregular, asymmetric DX-based polyhedra with variable edge lengths and vertices demonstrated experimentally, which significantly expands the set of possible 3D DNA-based assemblies that can be designed. ATHENA also enables external editing of sequences using caDNAno, demonstrated using asymmetric nanoscale positioning of gold nanoparticles, as well as providing atomic-level models for molecular dynamics, coarse-grained dynamics with oxDNA, and other computational chemistry simulation approaches.