§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.

Abstract. We show that the set of T-numbers in Mahler's classification of transcendental numbers supports a measure whose Fourier transform vanishes at infinity. A similar argument shows that U-numbers also support such a measure.

We give a transcendence measure of special values of functions satisfying certain functional equations. This improves an earlier result of Galochkin, and gives a better upper bound of the type for such a number as an S-number in the classification of transcendental numbers by Mahler.

In this paper we study the transcendence degree of fields generated over Q by the numbers associated with values of one-parameter subgroups of commutative algebraic groups. We show that in many instances these fields have a large transcendence degree when measured in terms of the available data.

Our method deals with points which are “well distributed” (in a sense which is made precise) among certain algebraic subgroups of the algebraic group under consideration. We verify that these results apply in many classical situations.

The problem of finding rational points on varieties defined by two additive cubic equations has attracted some interest. Davenport and Lewis [12], Cook [8] and Vaughan [16] showed that the pair of equations

with integer coefficients a,, bt always has a nontrivial solution when s = 18, s = 17, and 5 = 16 respectively. Vaughan's result in s = 16 variables is best possible since there are examples of pairs of equations (1) with s = 15 which fail to vanish simultaneously in the 7-adic field. However if the existence of a 7-adic solution is assured then Baker and Briidern [2], building on work of Cook [9], showed that s = 16 could be replaced by s = 15, and recently Briidern [5] has obtained the result with s = 14.

Let Q(x) = Q(x1,…, xn) є ęZ x1, …, xn] be a quadratic form. The primary purpose of this paper is to bound the smallest non-zero solution of the congruence Q(x) = 0 (mod q). The problem may be formulated as follows. We ask for the least bound Bn(q) such that, for any Ki > 0 satisfying

and any Q, the congruence has a non-zero solution satisfying

A version of Gauss's fifth proof of the quadratic reciprocity law is given which uses only the simplest group-theoretic considerations (dispensing even with Gauss's Lemma) and makes manifest that the reciprocity law is a simple consequence of the Chinese Remainder Theorem.

There are two types of quartic normal extensions of the rational field, depending on the Galois group of the generating equation. All such extensions are described here in a uniquely parametrized form.

The n-dimensional cross polytope, |x|+|x2|+…+|xn≤1, can be lattice packed with density δ satisfying

but proofs of this, such as the Minkowski-Hlawka theorem, do not actually provide such packings. That is, they are nonconstructive. Here we exhibit lattice packings whose density satisfies only

but by a highly constructive method. These are the densest constructive lattice packings of cross polytopes obtained so far.

Let J = (s1, s2, … ) be a collection of relatively prime integers, and suppose that π(n) = |J∩{1,2,…, n}| is a regularly varying function with index a satisfying 0 < α < l. We investigate the “stationary random sieve” generated by J, proving that the number of integers less than k which escape the action of the sieve has a probability mass function with approximate order k-α/2 in the limit as k → ∞. This result may be used to deduce certain asymptotic properties of the set of integers which are divisible by no s є J, in that it gives new information about the usual deterministic (that is, non-random) sieve. This work extends previous results valid when si=pi2, the square of the ith prime.

We speak of rigidity, if partial information about the prime decomposition in an extension of number fields K¦k determines the decomposition law completely (and hence the zeta function ζK), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

Let ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

We use a theorem of Loxton and van der Poorten to prove the transcendence of certain real numbers defined by digit patterns. Among the results we prove are the following. If k is an integer at least 2, P is any nonzero pattern of digits base k, and counts the number of occurrences (mod r) of p in the base k representation of n, then is transcendental except when k = 3, P = 1 and r = 2. When (r, k − 1) = 1 the linear span of the numbers has infinite dimension over Q, where P ranges over all patterns base k without leading zeros.