Let K be a number field of degree k > 1. We would like to know if a positive integer N can be represented as the sum, or the difference, of two norms of integral ideals of K. Suppose K/ℚ is abelian of conductor Δ. Then from the class field theory (Artin's reciprocity law) the norms are fully characterized by the residue classes modulo Δ. Precisely, a prime number p ∤ Δ (unramified in K) is a norm (splits completely in K), if, and only if,

where k is a subgroup of (ℤ/Δℤ)* of index k. Accordingly we may ask N to be represented as the sum

or the difference

of positive integers a, b each of which splits completely in K. For N to be represented in these ways the following congruences

must be solvable in α β є k, respectively. Moreover the condition

must hold. Presumably the above local conditions are sufficient for (−) to have infinitely many solutions and for (+) to have arbitrarily many solutions, provided N is sufficiently large in the latter case.

Let 1 ≤ M ≤ N − 1 be integers and K be a convex, symmetric set in Euclidean N-space. Associated with K and M, Mahler identified the Mth compound body of K, (K)m, in Euclidean (MN)-space. The compound body (K)M is describable as the convex hull of a certain subset of the Grassmann manifold in Euclidean (MN)-space determined by K and M. The sets K and (K)M are related by a number of well-known inequalities due to Mahler.

Here we generalize this theory to the geometry of numbers over the adèle ring of a number field and prove theorems which compare an adelic set with its adelic compound body. In addition, we include a comparison of the adelic compound body with the adelic polar body and prove an adelic general transfer principle which has implications to Diophantine approximation over number fields.

Suppose that is a distribution of N points in the unit square U = [0, 1]2. For every measurable set B in U, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2

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.

The best current bounds for the proportion of zeros of ζ(s) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ(s)M(s) near the -line, where

where μ(n) is the Möbius function, h(x) is a real polynomial with h(0) = 0, and y=Tθ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ(s) which are simple and on the critical line, then having the formula valid for 0 < θ < yields κ > 0·3562, having 0 < θ < gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 < θ < , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

Let s1, s2, … denote the squarefree numbers in ascending order. In [1], Erdős showed that, if 0 ≤ γ ≤ 2, then

where B(γ) is a function only of γ. In 1973 Hooley [4] improved the range of validity of this result to 0 ≤ γ ≤ 3, and then later gained a further slight improvement by a method he outlined at the International Number Theory Symposium at Stillwater, Oklahoma in 1984. We have, however, independently obtained the better improvement that (1) holds for

in contrast to the range

derived by Hooley. The main purpose of this paper is to substantiate our new result. Professor Hooley has informed me that there are similarities between our methods as well as significant differences.

Let

and A1, A2, …, An, a1, a2, …, an be positive integers. We consider the related integrals

and

Suppose that is a distribution of N points in U0, the closed disc of unit area and centred at the origin 0. For every measurable set B in ℝ2, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2

This paper presents new proofs of some classical transcendence theorems. We use real variable methods, and hence obtain only the real variable versions of the theorems we consider: the Hermite-Lindemann theorem, the Gelfond-Schneider theorem, and the Six Exponentials theorem. We do not appeal to the Siegel lemma to build auxiliary functions. Instead, the proof employs certain natural determinants formed by evaluating n functions at n points (alternants), and two mean value theorems for alternants. The first, due to Pólya, gives sufficient conditions for an alternant to be non-vanishing. The second, due to H. A. Schwarz, provides an upper bound.

Given ξ in [0, 1] let ξ = [0, a1, a2…] denote a simple continued fraction expansion of ξ Given the expansion ξ = [0, a1, a2, …] let

Thus with the conventions p1 = q0 = 1 and q−1 = p0 = 0 we have

Let K be an algebraic number field, [ K: ] = KΣ. Most of what we shall discuss is trivial when K = , so that we assume that K ≥ 2 from now onwards. To describe our results, we consider the classical device [2] of Minkowski, whereby K is embedded (diagonally-) into the direct product MK of its completions at its (inequivalent) infinite places. Thus MK is -algebra isomorphic to , and is to be regarded as a topological -algebra, dimRMK = K, in which K is everywhere dense, while the ring Zx of integers of K embeds as a discrete -submodule of rank K. Following the ideas implicit in Hecke's fundamental papers [6] we may measure the “spatial distribution” of points of MK (modulo units of κ) by means of a canonical projection onto a certain torus . The principal application of our main results (Theorems I–III described below) is to the study of the spatial distribution of the which have a fixed norm n = NK/Q(α). In §2 we shall show that, with suitable interpretations, for “typical” n (for which NK/Q(α) = n is soluble), these α have “almost uniform” spatial distribution under the canonical projection onto TK. Analogous questions have been considered by several authors (see, e.g., [5, 9, 14]), but in all cases, they have considered weighted averages over such n of a type which make it impossible to make useful statements for “typical” n.

Let k ≤ 2 be an integer, and set

Ek (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.

We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.

§1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.

The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equations

with 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writing

in which e(α) denotes e2πiα, we observe that

where Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).

Three differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

A well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.

A rational number is called a best approximant of the irrational number ζ if it lies closer to ζ than all rational numbers with a smaller denominator. Metrical properties of these best approximants are studied. The main tool is the two-dimensional ergodic system, underlying the continued fraction expansion.

§1. Introduction. In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t(–k) = t(k) be real,

where e(u) = e2πiu. Let p and p' denote primes, k an integer, and define

§1. Introduction. In 1985, Sárkõzy [11] proved a conjecture of Erdõs [2] by showing that the greatest square factor s(n)2 of the “middle” binomial coefficient satisfies for arbitrary ε > 0 and sufficiently large n

Where

Dörner [8] has recently proved a result for systems of additive equations over algebraic number fields, which we restate in a more specific setting.