We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of ‘universal covers’ of rigid divisible commutative finite Morley rank groups.

If $3\leqslant n<\unicode[STIX]{x1D714}$ and $V$ is a vector space over $\mathbb{Q}$ with $|V|\leqslant \aleph _{n-2}$ , then there is a well ordering of $V$ such that every vector is the last element of only finitely many length- $n$ arithmetic progressions ( $n$ -APs). This implies that there is a set mapping $f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$ with no free set which is an $n$ -AP. If, however, $|V|\geqslant \aleph _{n-1}$ , then for every set mapping $f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$ there is a free set which is an $n$ -AP.

We generalize Skriganov’s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov’s celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erdős, and others. Finally, we use o-minimality to describe large classes of sets to which our counting results apply.

In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

The purpose of this paper twofold. Firstly, we establish $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if $\unicode[STIX]{x1D707}$ is a self-similar measure satisfying the strong separation condition and $\unicode[STIX]{x1D707}$ is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of $\unicode[STIX]{x1D707}$ defined by

$$\begin{eqnarray}\bigg\{x\in \mathbb{R}^{d}\,\bigg|\,\lim _{r{\searrow}0}\frac{\log \unicode[STIX]{x1D707}(B(x,r))}{\log r}=\unicode[STIX]{x1D6FC}\bigg\}\end{eqnarray}$$

$\unicode[STIX]{x1D6F1}_{3}^{0}$

We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski–Kazhdan style motivic integration for certain types of nonarchimedean $o$ -minimal fields, namely power-bounded $T$ -convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the $\text{VF}$ -sort and the $\text{RV}$ -sort of the language ${\mathcal{L}}_{T\text{RV}}$ . Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to $\unicode[STIX]{x2124}$ . The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction.

Measuring says that for every sequence ${\left( {{C_\delta }} \right)_{\delta < {\omega _1}}}$ with each ${C_\delta }$ being a closed subset of δ there is a club $C \subseteq {\omega _1}$ such that for every $\delta \in C$, a tail of $C\mathop \cap \nolimits \delta$ is either contained in or disjoint from ${C_\delta }$. We answer a question of Justin Moore by building a forcing extension satisfying measuring together with ${2^{{\aleph _0}}} > {\aleph _2}$. The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable structures as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:{\omega _1} \to {\omega _1}$ mod. countable at each stage.

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^{r}$ -smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a $C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height $H$ in a compact piece of a complex-analytic set of dimension $k$ in $\mathbb{C}^{m}$ are contained in $O(1)$ complex-algebraic hypersurfaces of degree $(\log H)^{k/(m-k)}$ . This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.

Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$ , we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$ -closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$ -closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$ , the lattice of ${\mathcal{U}}$ -ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$ -distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$ . From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$ , then it has an $(\unicode[STIX]{x1D714},3)$ -representation.

A new generalized class of fuzzy implications, called ( $h,f,g$ )-implications, is introduced and discussed in this paper. The results show that the new fuzzy implications possess some good properties, such as the left neutrality property and the exchange principle.

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$ -tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$ ) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

We study the relationship between a $\unicode[STIX]{x1D705}$ -Souslin tree $T$ and its reduced powers $T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$ .

Previous works addressed this problem from the viewpoint of a single power $\unicode[STIX]{x1D703}$ , whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph _{6}$ -Souslin tree $T$ and a sequence of uniform ultrafilters $\langle {\mathcal{U}}_{n}\mid n<6\rangle$ such that $T^{\aleph _{n}}/{\mathcal{U}}_{n}$ is $\aleph _{6}$ -Aronszajn if and only if $n<6$ is not a prime number.

This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.

We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$ , the set of all subsets of a set $A$ , onto $A$ , then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$ . In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$ , provable without AC.

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.

Let $L$ be a countable language. We say that a countable infinite $L$ -structure ${\mathcal{M}}$ admits an invariant measure when there is a probability measure on the space of $L$ -structures with the same underlying set as ${\mathcal{M}}$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of ${\mathcal{M}}$ . We show that ${\mathcal{M}}$ admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in $\text{Aut}({\mathcal{M}})$ of an arbitrary finite tuple of ${\mathcal{M}}$ fixes no additional points. When ${\mathcal{M}}$ is a Fraïssé limit in a relational language, this amounts to requiring that the age of ${\mathcal{M}}$ have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$ . We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$ . The reductions are definable in theory $S_{2}^{1}$ . We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.