In [1], [2] Besicovitch showed that it is possible to translate each straight line in the plane so that the union of all the translates has zero plane measure. More recently Besicovitch and Rado [3] and independently Kinney [12] showed that the same can be done with arcs of circles instead of straight lines (see also Davies [6]). Allowing rotations as well as translations, Ward [18] showed that all plane polygonal curves can be “packed” thus (allowing overlapping) into zero plane measure, and then Davies [7], making use of Besicovitch's construction, showed translations alone to be sufficient, although these papers in fact contained stronger results concerning Hausdorff measure; the results were further generalized in [16]. The question has naturally been asked whether the class of all plane rectifiable curves can be packed by isometries (translations and rotations) into zero plane measure, but a special case of the main theorem of the present paper shows that this is impossible. The corresponding question remains open for the much smaller class of algebraic curves, or even conies.

We show that a complete metric space X has an essentially unique “nice” zero-dimensional dense Gδ subset, and derive from this a complete algebraic description of the “category algebra” (of Borel modulo first category sets) of X.

A generalization and simplification of F. John's theorem on ellipsoids of minimum volume is proven. An application shows that for 1 ≤ p < 2, there is a subspace E of Lp and a λ > 1 such that 1E has no λ-unconditional decomposition in terms of rank one operators.

If C is a convex open set of a Hausdorff locally convex space, then for every compact set K ⊂ C, the closed convex hull of K is contained in C. It follows that the same is true if C is a countable intersection of convex open sets of E. However, as it is well known (and also follows from §7(a)) a convex of Gδ is not always a countable intersection of open convex sets. This paper is devoted to showing that the Gδ convex sets have some remarkable properties.

We are concerned with invertible transformations of the unit n-dimensional cube In, 2 ≤ n ≤ ∞, which preserve n-dimensional Lebesgue measure μ. Following Halmos [4], we denote the space of all such transformations by G = G(In), and the subset of G consisting of homeomorphisms by M = M(In). We ask to what extent, and in what sense, can we approximate an arbitrary transformation g in G by a homeomorphism h in M. New results are obtained in the course of presenting a new proof of the theorem of J. Oxtoby and H. E. White, Jr., stated below.

In this note we remark on certain “universal fixed messages” for the Rivest-Shamir-Adleman cryptosystem [1], and we describe how in certain cases these universal fixed messages can play a role in an attempt to break the cryptosystem. This use of the universal fixed messages in an attempt to break the RSA cryptosystem† involves a certain prime factorization technique using the fixed messages. We characterize the situation in which such an attempt would be successful in practice, and we show that a user of the RSA cryptosystem can easily arrange to avoid this situation. Hence the use of the present technique does not pose a threat to the security of the RSA cryptosystem.

Suppose that we have a system of congruences ai (mod ni) 1 < n1 < … < ni < … < nk such that every integer is congruent to at least one ai (mod ni), then we say that it is a covering system of congruences. If ni | m, 1 ≤ i ≤ k, we say that m is a covering number. We shall use the symbol ℕ to denote the natural numbers together with zero, then m is a covering number if, for each q there is an aq such that

Barban and Vehov posed in [1] the problem of minimizing the quadratic form

under the conditions

where 1 ≤ z1 < z2 and μ(n) is the Möbius function. They also commented on the connections of this problem with zero-density estimates of L-functions near the lines δ = 1 and ½, and with Linnik's prime number theorem. This program was carried out some years later by other authors, in the first place by Selberg [10], and later by Motohashi ([7]–[9]), Graham [2] and the present author ([4], [5]).

Let f = f(x, y) be a quadratic form with real coefficients in two integer variables x, y. Let V(f) be the set of values taken by f(x, y) at points (x, y) ≠ (0,0). Impose the same conditions on a second form f′. Trivially, f equivalent to f′ implies V(f) = V(f′). It will be shown that the converse implication holds in general for definite forms; the obvious exception f = x2 + xy + y2, f′ = x2 + 3y2 will be shown to be essentially the only one.

This paper originated with the observation that while all of the known stable lattice packings of spheres are highly symmetric, it is futile to try to prove a converse statement: the ordinary integer-lattice provides a distinctly unstable packing of spheres, but admits a large group of orthogonal symmetries nonetheless. The integerlattice is in fact very unstable—the slightest perturbation places the spheres in a more efficient configuration. We will call such a lattice fragile. The purpose of this note is to prove that a highly symmetric lattice must be either stable or fragile.

Let K be any finite (possibly trivial) extension of ℚ, the field of rational numbers. Let denote the ring of integers of K, and let M ⊆ be a full module in K thus a free ℤ-module of rank [K : ℚ] contained in ; ℤ denoting the ring of rational integers. Regarding as an abelian group, the index (: M) is finite. Suppose that m1, …, mk is a ℤ-basis for M and let a ∊ Then the polynomial

(the xi; being indeterminates) will be called a full-norm polynomial; here NK/ℚ denotes the norm mapping from K to ℚ. Apart from constant factors, such a polynomial f(x) is necessarily irreducible in ℤ[x].

G. Higman [5] first considered conditions on a group G sufficient to ensure that for any ring R with no zero-divisors the group-ring RG contains no zero-divisors. It has been shown by various authors that if G belongs to one of the classes of locally indicible groups [5], right-ordered groups [6], polycyclic groups [4] or positive one-relator groups [1] then it is enough that G should be torsionfree. The proofs rely heavily on the special properties of the classes of groups involved but it may be conjectured that it is a sufficient condition in general that G should be torsionfree and no counterexamples are known.

If G is a topological group then we can think of G acting on itself by multiplying on the left. We would like to know when this action has the property that whenever g and h are distinct elements of G, then the element xg does not get arbitrarily close to xh as x varies in G. It is natural to say that this is the case if {(xg, xh): x∈G} is separated from the diagonal of G × G by a uniform neighbourhood of the diagonal.

Lyndon's axiomatic methods are used in [1] to show, among other things, that a group G with an integer valued length function satisfying certain conditions is free. At the end of his paper [2] Lyndon gives a method of embedding such a group in a free group whose natural length function extends the function on G. We construct here a simpler embedding with the same property.

The Picard group P(ZG) of the integral group ring ZG is defined as the class group of two-sided invertible ZG-ideals of QG modulo those principal ideals generated by an invertible central element. The basic properties of Picard groups have been established by A. Fröhlich, I. Reiner and S. Ullom [1], [2], [3]. In this note we settle an outstanding question by exhibiting a class of finite p-groups G whose Picard groups contain nontrivial elements which are represented by principal ideals; these elements remain nontrivial in P(ZpG) also. We obtain these ideals from outer automorphisms of the groups.

We say a motion g brings a mobile convex body K into inner contact with a fixed body K0 if the image gK lies in K0 and shares a boundary point with K0; we speak of the inner contact being at the common boundary point. The mobile body K is said to roll freely in K0 if, corresponding to each boundary point x of K0 and each rotation R, there is a translation t such that RK + t = gK has inner contact with K0 at x.

In an article generalising work of Roquette and Zassenhaus, Connell and Sussman [2] have demonstrated the importance of certain prime ideals in a number field k0 for estimating the l-rank of the class group of an extension k. These ideals have a power prime to l which is principal and all their prime factors in k have ramification index divisible by l. The products of the prime divisors of these ideals in the normal closure K of k/k0 are invariant under Gal (k/k0). Thus certain roots in k of the ideals in k0 are in some sense fixed by the Galois group. This leads to the concept of ambiguous ideals in an extension k/k0 which is not necessarily normal.

A Borel isomorphism that, together with its inverse, maps ℱσ-sets to ℱσ-sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism, for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.