We show that every continuous simple curve with σ-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and study the validity of the results in higher dimension.

A limit of rational varieties need not be rational, even when all varieties in the family are projective and have at most terminal singularities.

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

Let G be a connected perfect real Lie group. We show that there exists α < dim G and p ∈ $\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure on G, then the pth convolution power μ*p is absolutely continuous with respect to the Haar measure on G, with arbitrarily smooth density. As an application, we obtain that if A ⊂ G is a Borel set with Hausdorff dimension at least α, then the p-fold product set Ap contains a non-empty open set.

Among monoidal categories with finite coproducts preserved by tensoring on the left, we characterise those with finite biproducts as being precisely those in which the initial object and the coproduct of the unit with itself admit right duals. This generalises Houston's result that any compact closed category with finite coproducts admits biproducts.

We generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extension L/$\mathbb{Q}$ exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modular L-function L(s, f), the fundamental discriminants d for which the d-quadratic twist of L(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

A celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

Recent workers [1, 3] have proved density theorems about the rational points on K3 surfaces of the form

$$ V:\;X_0^4+cX_1^4=X_2^4+cX_3^4 $$

In this paper we study the following question: given a semigroup ${\mathcal S}$ of partial isometries acting on a complex separable Hilbert space, when does the selfadjoint semigroup ${\mathcal T}$ generated by ${\mathcal S}$ again consist of partial isometries? It has been shown by Bernik, Marcoux, Popov and Radjavi that the answer is positive if the von Neumann algebra generated by the initial and final projections corresponding to the members of ${\mathcal S}$ is abelian and has finite multiplicity. In this paper we study the remaining case of when this von Neumann algebra has infinite multiplicity and show that, in a sense, the answer in this case is generically negative.

In this paper, we prove a Tb theorem on product spaces $\mathbb{R}$n × $\mathbb{R}$m, where b(x1, x2) = b1(x1)b2(x2), b1 and b2 are para-accretive functions on $\mathbb{R}$n and $\mathbb{R}$m, respectively.

By considering the Bredon analogue of complete cohomology of a group, we show that every group in the class $\cll\clh^{\mathfrak F}{\mathfrak F}$ of type Bredon-FP∞ admits a finite dimensional model for $E_{\frak F}G$.

We also show that abelian-by-infinite cyclic groups admit a 3-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for $\mathfrak {F}$, the class of virtually cyclic groups, the class of $\cll\clh^{\mathfrak F}{\mathfrak F}$-groups contains all locally virtually soluble groups and all linear groups over ${\mathbb{C}}$ of integral characteristic.

Assuming CH, the continuum hypothesis, holds we show, by completing an attack first discovered by Roy Davies, that for each α between 0 and 1 there is a subring, in fact a subfield, of R with Hausdorff dimension α.

The category of rational SO(2)–equivariant spectra admits an algebraic model. That is, there is an abelian category ${\mathcal A}$(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)–equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra?

The category ${\mathcal A}$(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non–Noetherian ring. As a consequence, the standard techniques for constructing a monoidal model structure cannot be applied. In this paper a monoidal model structure on ${\mathcal A}$(SO(2)) is constructed and the derived tensor product on the homotopy category is shown to be compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K(p)–local stable homotopy category.

A monoidal Quillen equivalence to a simpler monoidal model category R•-mod that has explicit generating sets is also given. Having monoidal model structures on ${\mathcal A}$(SO(2)) and R•-mod removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)–equivariant spectra.