In this paper we shall be concerned with the growth of functions in the class ℳp, where we write f ∈ ℳp 1 <p<∞, if:

(i) f is absolutely continuous on bounded subintervals of [0, ∞]);

(ii) f(0) = 0; and

(iii) .

The main object of this note is to prove that in three-space the sausage arrangement is the densest packing of four unit balls. Our method can be used to determine minimal arrangements with respect to various properties of four-ball packings, as we point out in Section 3.

We shall say that the sets A, B ⊂ Rk are equivalent, if they are equidecomposable using translations; that is, if there are finite decompositions and vectors x1,…, xd∈Rk such that Bj = Aj + xj, (j = 1,…,d). We shall denote this fact by In [3], Theorem 3 we proved that if A ⊂ Rk is a bounded measurable set of positive measure then A is equivalent to a cube provided that Δ(δA)<k where δA denotes the boundary of A and Δ(E) denotes the packing dimension (or box dimension or upper entropy index) of the bounded set E. This implies, in particular, that any bounded convex set of positive measure is equivalent to a cube. C. A. Rogers asked whether or not the set

Let |θ| < π/2 and . By refining Selberg's method, we study the large values of as t → ∞ For σ close to ½ we obtain Ω+ estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that

and

Our results supplement those of Montgomery which are good when σ > ½ is fixed.

General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ + Mδc, in terms of those of the measures dμ and (x − c)2dμ. In particular, these relations allow us to show that Nevai's class M(0, 1) is closed under adding a mass point, as well as obtain several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

In this paper we characterize Fountain-Gould left orders in abelian regular rings. Our first approach is via the multiplicative semigroups of the rings. We then represent certain rings by sheaves. Such representations lead us to a characterization of left orders in abelian regular rings such that all the idempotents of the quotient ring lie in the left order.

In Mahler's classification of complex numbers [10] (see [4]), a transcendental number ξ is called a U-number if there exists a fixed integer N ≥ 1 so that for all ω > 0, there exists a polynomial so that

where the height h(f) = max {|α0|, |α1|, …, |αN|}. The number ξ is called a Um-number if the above holds for N = m but for no smaller value of N (examples and further details may be found in [9,1 and 2]). Thus the set of U1-numbers is precisely the set of Liouville numbers. In this paper we investigate the statistical behavior of the partial quotients of real U-numbers, in particular, U2-numbers. In addition, we demonstrate the existence of a U2-number with the property that if it is translated by any nonnegative integer and then squared, the result is a Liouville number. Related results involving badly approximate U2-numbers are also discussed.

Let ℳ denote the class of all finite positive Borel measures μ in Rn with compact support. We define the Fourier transform of μεℳ by writing

and the α-energy of μ is given by

It is shown that a convex body is determined uniquely among all convex bodies by the volumes of its projections onto all hyperplanes through the origin if and only if it is a parallelotope.

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.

In this note we give a direct algebraic proof of a theorem of Warfield on algebraically compact modules. It is shorter than the one given by Azumaya in [1], in that it does not use the embedding of a module M into M** (where M* is the character Homz (M, Q/Z)).

It is proved that for a symmetric convex body K in ℝn, if for some τ >0, |K ⋂ (x + τK) depends on ‖x‖K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies are studied.