The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

Let P be a finitely generated ideal of a commutative ring R. Krull's principal ideal theorem states that if R is Noetherian and P is minimal over a principal ideal of R, then P has height at most one. Straightforward examples show that this assertion fails if R is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces.

Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.

This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.

We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.

In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$.