A permutation group G on a set Ω is said to be binary if, for every n ∈ ℕ and for every I, J ∈ Ωn, the n-tuples I and J are in the same G-orbit if and only if every pair of entries from I is in the same G-orbit to the corresponding pair from J. This notion arises from the investigation of the relational complexity of finite homogeneous structures.

Cherlin has conjectured that the only finite primitive binary permutation groups are the symmetric groups Sym(n) with their natural action, the groups of prime order, and the affine groups V ⋊ O(V) where V is a vector space endowed with an anisotropic quadratic form.

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to PSL2(q), 2B2(q), 2G2(q) or PSU3(q). Our method uses the notion of a “strongly non-binary action”.

We construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

A. Avilés and C. Brech proved an intriguing result about the existence and uniqueness of certain injective Boolean algebras or Banach spaces. Their result refines the standard existence and uniqueness of saturated models. They express a wish to obtain a unified approach in the context of category theory. We provide this in the framework of weak factorisation systems. Our basic tool is the fat small object argument.

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.

We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.

We provide the first explicit example of a universal Hilbert set ${\Ncal S}$ having asymptotic density 1 in the set of integers. More precisely, the number of integers not in ${\Ncal S}$ with absolute value ≤ X is bounded by X/(log X)δ, where δ = 1 − (1 + loglog 2)/(log 2) = 0.086071. . ..

Let K be a field of characteristic zero and let R = K[X1, . . .,Xn], with standard grading. Let ${\mathfrak m}$ = (X1, . . ., Xn) and let E be the *injective hull of R/${\mathfrak m}$. Let An(K) be the nth Weyl algebra over K. Let I, J be homogeneous ideals in R. Fix i, j ≥ 0 and set M = HiI(R) and N = HjJ(R) considered as left An(K)-modules. We show the following two results for which no analogous result is known in charactersitc p > 0.

1. (i) $H^l_{\mathfrak m}$(TorRν(M, N)) ≅ E(n)al,ν for some al,ν ≥ 0.

2. (ii) For all ν ≥ 0; the finite dimensional vector space TorAn(K)ν(M♯, N) is concentrated in degree -n (here M♯ is the standard right An(K)-module associated to M).

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.

Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.

The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

Let A ⊂ ℝ be finite. We quantitatively improve the Balog–Wooley decomposition, that is A can be partitioned into sets B and C such that

$ \begin{equation*} \max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}. \end{equation*} $

$ \begin{equation*} |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}. \end{equation*} $