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.

In this paper the power values of the sum of factorials and a special diophantine problem related to the Ramanujan-Nagell equation are studied. The proofs are based on deep analytic results and Baker's method.

We generalise the approximation theory described in Mahier's paper “Perfect Systems” to linked simultaneous approximations and prove the existence of nonsingular approximation and of transfer matrices by generalising Coates' normality zig-zag theorem. The theory sketched here may have application to constructions important in the theory of diophantine approximation.

Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

This paper is the first part of a long delayed revision of the manuscript ‘The growth conditions recurrence sequences’ (circulated in 1982) in which the authors outlined a proof of the now well known theorem on the finiteness of the number of solutions of S-unit equations. The argument lifting the result from number fields to arbitrary fields of characteristic zero has original features.

Let K be an algebraic number field, [K: Q] = κ є N; only the case κ > 1 is of interest in this paper. Let f be any non-zero ideal in ZK, the ring of integers of K, and let b be any ray-class (modx f) of K. In this paper we answer a question of P. Erdös (private communication) about the “maximum-growth-rate” of the functions

and

the sum here taken over all ray-classes (modx f), while N(a) is the absolute norm of a. Let

and

where, as usual, for x є R, log+ x - log max {1, x}. We prove

THEOREM 1. For every f, b we have

The best lattice quantizers seem to be duals of extreme lattices. The quantizing constant associated with the dual lattice of Barnes's senary form φ6 is found, together with a new type of quantizing technique. The quantizing constant is better than expected in the sense that it is better than D*6 even though D6 provides a denser packing. This is the smallest dimension for which this occurs.

Among all possible semiregular continued fraction expansions of an irrational number the one with the best approximation properties, in a well-defined and natural sense, is determined. Some properties of this so called optimal continued fraction expansion are described.

We improve W. Schmidt's lower bound for the slice (intersection of two halfspheres) discrepancy of point distributions on spheres and show that this estimate is up to a logarithmic factor best possible. It is shown that the slice and spherical cap discrepancies are equivalent for the definition of uniformly distributed sequences on spheres.

One may perhaps doubt whether in the geometry of numbers any particular family of lattices deserves such an attention as, for example, BCH codes receive in coding theory. However, only recently a quite interesting family has emerged. The general case of these lattices considered by Rosenbloom and Tsfasman [5, Section 2] parallels Goppa's construction of codes from algebraic curves. Here we shall take a closer look at the case of genus zero where some special features of Goppa's early codes will show up again: There is a lattice Λ (L, g) in n-dimensional euclidean space associated with a subset L of the field , and a polynomial g satisfying g(є) ≠ for all λ є L. For g = zd previously known sphere packings are recovered and generalized. A nonconstructive argument shows that for n → ∞ and some irreducible polynomials g Minkowski's lower packing bound is met (this being not achieved in [5] where q is fixed, but the genus grows; cf. also [4]).