We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski's theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

Given a ±1 random completely multiplicative function f, we prove by estimating moments that the limiting distribution of the normalized sumconverges to the standard Gaussian distribution as x → ∞ when r restricts summation to n having o(log log log x) prime factors. We also give an upper bound for the large deviations ofwith the sum restricted to numbers having a fixed number k of prime factors.

Somos 4 sequences are sequences of numbers defined by a bilinear recurrence relation of order 4 and include elliptic divisibility sequences as a special case. In this paper, we describe valuations of Somos 4 sequences in terms of canonical local heights on the associated elliptic curves. We consider both Archimedean and non-Archimedean valuations. As applications, we study the asymptotic behaviour of valuations of Somos 4 sequences and obtain another proof of the integrality of certain Somos 4 sequences.

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3 and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.

We establish that the set of pairs (α, β) of real numbers such thatwhere ‖ · ‖ denotes the distance to the nearest integer, has full Hausdorff dimension in R2. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems.

The main purposes of this paper are to establish and exploit the result that, over a complete (Noetherian) local ring R of prime characteristic for which the Frobenius homomorphism f is finite, the appropriate restrictions of the Matlis-duality functor provide an equivalence between the category of left modules over the Frobenius skew polynomial ring R[x, f] that are Artinian as R-modules and the category of right R[x, f]-modules that are Noetherian as R-modules.

We show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.

We define the symmetric Auslander category As(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic complexes of projective modules.

The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that As(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categorieswhere GMor(R) is the stable category of Gorenstein projective objects in the abelian category Mor(R) of homomorphisms of R-modules.

This result is set in the wider context of a theory for As(R) and Bs(R), the symmetric Bass category which is defined dually.

We consider generalisations of the so-called Euler adic and investigate dynamical properties like ergodicity and total ergodicity. We prove the existence of a unique fully-supported ergodic measure for these generalisations. We also investigate the structure of non-fully-supported ergodic measures and in addition show that each of these measures (fully- and non-fully-supported) is also totally ergodic. In order to determine these dynamical properties we find closed-form expressions for the generalised Eulerian numbers. Additionally we extend a result given by Frick and Petersen to a wider class of adic transformations.

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

Let K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

We consider the problem of finding the maximum possible size of a family of d-dimensional subcubes of the n-cube {0, 1}n, none of which is contained in the union of the others. (We call such a family ‘irredundant’). Aharoni and Holzman [1] conjectured that for d > n/2, the answer is (which is attained by the family of all d-subcubes containing a fixed point). We prove this in the case where all the subcubes go through either (0, 0, . . . , 0) or (1, 1, . . . , 1). We also give a new proof of an upper bound of Meshulam [6]. Finally, we consider the case d ≤ n/2. We give a probabilistic construction showing that in this case, Meshulam's upper bound is correct to within a constant factor. When n = 2d, we construct an irredundant family of size .

We give a framework for combining n monads on the same category via distributive laws satisfying Yang–Baxter equations, extending the classical result of Beck which combines two monads via one distributive law. We show that this corresponds to iterating n-times the process of taking the 2-category of monads in a 2-category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict n-category monad on n-dimensional globular sets; we first construct for each i a monad for composition along bounding i-cells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary Yang–Baxter equations.

L'aspect fractal de la suite des diviseurs d'un entier normal peut être révélé de diverses manières, comme l'existence d'une dimension de Hausdorff convenablement définie [5]. Cependant, les critères les plus fins sont liés à la conjecture d'Erdős datant de la fin des années 1930, mentionnée dans [3] et établie dans [6], selon laquelle presque tout entier possède deux diviseurs dont le rapport est dans l'intervalle ]1,2[.

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple.

Two appendices introduce results and examples around the concept of quasi-product.

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

This paper determines which Stiefel–Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F2-vector space of Stiefel–Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP3 bundles. These Stiefel–Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of 3 bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

Legendrian contact homology (LCH) is a powerful non-classical invariant of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and non-commutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products – and, more generally, an A∞ structure – on the linearized LCH. We apply the products and A∞ structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

If Γ is a discrete subgroup of PSL(3, ℂ), it is determined the equicontinuity region Eq(Γ) of the natural action of Γ on ℙ2ℂ. It is also proved that the action restricted to Eq(Γ) is discontinuous, and Eq(Γ) agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of Γ in the sense of Kulkarni, Λ(Γ), contains at least three complex lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which Γ acts discontinuously. Moreover, if Λ(Γ) contains at least four complex lines and Γ acts on ℙ2ℂ without fixed points nor invariant complex lines, then each connected component of Eq(Γ) is a holomorphy domain and a complete Kobayashi hyperbolic space.