Let Ω be an algebraic number field, and let NΩ ⊂ ℚ be the group of norms of fractional ideals of Ω. Then NΩ is a subgroup of the positive rationals; the latter is the direct sum of a denumerable infinity of infinite cyclic groups, and so it is free abelian; thus NΩ is free abelian, and, since it is not finitely generated, we must have qua abstract groups. The purpose of this paper is, in the first place, to find a “metrical ” way of distinguishing these isomorphic groups, and, to this end, we introduce the notion of Farey density, defined as follows; let X be a positive integer, and consider the Farey section ℱ(X) of order X, thus the set of all reduced positive fractions with denominator < X; then the quotient

measures the proportion of elements of ℱ(X) which are in NΩ, and, as X → ∞, it gives a measure of the “density ” of fractional ideal norms in the rational interval (0, 1).

Given $n\in \mathbb{N}$ and $\unicode[STIX]{x1D70F}>1/n$, let ${\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ denote the classical set of $\unicode[STIX]{x1D70F}$-approximable points in $\mathbb{R}^{n}$, which consists of $\mathbf{x}\in \mathbb{R}^{n}$ that lie within distance $q^{-\unicode[STIX]{x1D70F}-1}$ from the lattice $(1/q)\mathbb{Z}^{n}$ for infinitely many $q\in \mathbb{N}$. In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}>1/n$ almost all points on ${\mathcal{M}}$ are not $\unicode[STIX]{x1D70F}$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set ${\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$. In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], which enables us to find a sharp lower bound for $\dim {\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ for any $C^{2}$ submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}$ satisfying $1/n\leqslant \unicode[STIX]{x1D70F}<1/m$. Here $m$ is the codimension of ${\mathcal{M}}$. We also show that the condition on $\unicode[STIX]{x1D70F}$ is best possible and extend the result to general approximating functions.

It is well known that a tournament (complete oriented graph) on n vertices has at most directed triangles, and that the constant is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some “higher order” versions of this statement. For example, if we give each 3-set from an n-set a cyclic ordering, then what is the greatest number of “directed 4-sets” we can have? We give an asymptotically best possible answer to this question, and give bounds in the general case when we orient each d-set from an n-set.

This second part follows on directly from the first part that appears on pages 125–156 of this volume. The references for this part are included amongst the references appearing at the end of the first part.

Let $P^{+}(n)$ denote the largest prime factor of the integer $n$ and $P_{y}^{+}(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, first we show that the triple consecutive integers with the two patterns $P^{+}(n-1)>P^{+}(n)<P^{+}(n+1)$ and $P^{+}(n-1)<P^{+}(n)>P^{+}(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any$J\in \mathbb{Z}$, $J\geqslant 3$, the $J$-tuple consecutive integers with the two patterns $P^{+}(n+j_{0})=\min _{0\leqslant j\leqslant J-1}P^{+}(n+j)$ and $P^{+}(n+j_{0})=\max _{0\leqslant j\leqslant J-1}P^{+}(n+j)$ also have a positive proportion, respectively. Second, for $y=x^{\unicode[STIX]{x1D703}}$ with $0<\unicode[STIX]{x1D703}\leqslant 1$ we show that there exists a positive proportion of integers $n$ such that $P_{y}^{+}(n)<P_{y}^{+}(n+1)$. Specifically, we can prove that the proportion of integers $n$ such that $P^{+}(n)<P^{+}(n+1)$ is larger than 0.1356, which improves the previous result “0.1063” of the author.

Let M be a compact flat Riemannian manifold of dimension n, and Γ its fundamental group. Then we have the following exact sequence (see [1])

where Zn is a maximal abelian subgroup of Γ and G is a finite group isomorphic to the holonomy group of M. We shall call Γ a Bieberbach group. Let T be a flat torus, and let Ggr act via isometries on T; then ┌ acts isometrically on × T where is the universal covering of M and yields a flat Riemannian structure on ( × T)/Γ. A flat-toral extension (see [9, p. 371]) of the Riemannian manifold M is any Riemannian manifold isometric to ( × T)/Γ where T is a flat torus on which Γ acts via isometries. It is convenient to adopt the convention that a single point is a 0-dimensional flat torus. If this is done, M is itself among the flat toral extensions of M. Roughly speaking, this is a way of putting together a compact flat manifold and a flat torus to make a new flat manifold the dimension of which is the sum of the dimensions of its constituents. It is, more precisely, a fibre bundle over the flat manifold with a flat torus as fibre.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

In a recent paper of Ellenberg, Oberlin, and Tao [The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), 1–25], the authors asked whether there are Besicovitch phenomena in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set of measure zero in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. Furthermore, we prove that any Kakeya set in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{2} $ or ${ \mathbb{Z} }_{p}^{2} $ is of Minkowski dimension 2.

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the $d$-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^{p}$ estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.

Given a commutative semigroup (S, +) with identity 0 and u × v matrices A and B with nonnegative integers as entries, we show that if C = A – B satisfies Rado's columns condition over ℤ, then any central set in S contains solutions to the system of equations . In particular, the system of equations is then partition regular. Restricting our attention to the multiplicative semigroup of positive integers (so that coefficients become exponents) we show that the columns condition over ℤ is also necessary for the existence of solutions in any central set (while the distinct notion of the columns condition over Q is necessary and sufficient for partition regularity over ℕ\{1}).

This paper gives an algorithm for generating all the solutions in integers x0, x1…, xn of the inequality

where 1, α1, …,αn are numbers, linearly independent over the rationals, in a real algebraic number field of degree n + 1 ≥ 3 and c is any sufficiently large positive constant. It is well known [2, p. 79] that if c is small enough, then (1) has no integer solutions with x1…, xn not all zero.

Let p(n) be the usual partition function. Let l be an odd prime, and let r (mod t) be any arithmetic progression. If there exists an integer n ≡ r (mod t) such that p(n) ≢ 0 (mod l), then, for large X,

We consider the differential equation

where w(x) = xα for α > -1, q is a real-valued member of (0, ∞) and λ is a complex number with

We study the minimal length of faithful nuclear representations of operators acting between finite-dimensional Banach spaces and the related minimal number of contact points of the John ellipsoid of maximal volume contained in the unit ball of a finite-dimensional Banach space. In both cases the classical upper estimates, which follow from the Caratheodory theorem, are shown to be exact. Related isometric characterizations of are proved.

We obtain explicit lower bounds on the lattice packing densities δL of superballs G of quite a general nature, and we conjecture that as the dimension n approaches infinity, the bounds are asymptotically exact. If the conjecture were true, it would follow that the maximum lattice-packing density of the Iσ-ball is 2−n(1+σ(1)) for each σ in the interval 1 ≤ σ ≤ 2.

Let (bn) be a sequence of integers, obtained by traversing the rows of Pascal's triangle, as follows: start from the element at the top of the triangle, and at each stage continue from the current element to one of the elements at the next row, either the one immediately to the left of the current element or the one immediately to its right. Consider the distribution of the sequence (bnα) modulo 1 for an irrational α. The main results show that this sequence “often” fails to be uniformly distributed modulo 1, and provide answers to some questions raised by Adams and Petersen.

We examine the asymptotic form of a fundamental set of solutions of the third-order equation

as x → ∞, where the leading coefficient q is nowhere zero in some interval [a, ∞). The equation is self-adjoint in the case when p, r and iq are real. However, our analysis is not confined to this case, and we generally take the coefficients to be complex-valued.

We begin by denning the notion of a tangential limit for a function f denned in the unit disc

Let be a positive continuous function on (0, 1) for which

Suppose B>0, -, and define

where The region . makes tangential contact with the boundary U of the unit disc at ei; when (r) = ( l - r2), for instance, (, , 1) is the disc with radius and centre ei

In [8,9] Jayne and Rogers studied piece-wise closed maps and ℱσ maps between metric spaces. A map f of a metric space X into a metric space Y is said to be an ℱσ map if: (a) f maps ℱσ-sets in X to ℱσ-sets in Y; and (b) f1 maps ℱσ-sets in Y back to ℱσ-sets in X. A map fof a metric space X into a metric space Y is said to be piece-wise closed if:it is possible to find a sequence X1, X2,… of closed sets in X, with with each setf(Xi), i ≥ 1, closed in Y, and with the restriction offto each Xi, a closed map (i.e., a continuous map that maps closed sets to closed sets).

We prove that among all flag triangulations of manifolds of odd dimension $2r-1$ , with a sufficient number of vertices, the unique maximizer of the entries of the $f$ -, $h$ -, $g$ - and $\unicode[STIX]{x1D6FE}$ -vector is the balanced join of $r$ cycles. Our proof uses methods from extremal graph theory.