Throughout this note the letters D and K denote a commutative integral domain with 1 and its field of fractions.

In a series of papers we have analysed the embedding of certain groups H as normal subgroups of absolutely irreducible skew linear groups G, see [7], [8], [9] and [10]. Here we drop the absolutely irreducibility assumption on G. If H is locally finite we derive relatively strong conditions on G, although not as strong as when G is absolutely irreducible. If H is abelian, very much in contrast to the absolutely irreducible case, we show that nothing can be said. The phrase “bounded by an integer-valued function of n only” we abbreviate to “n-bounded”.

We continue the study of approximate subdifferentials initiated in [3], this time for functions on arbitrary locally convex spaces. The complexity of the infinite dimensional theory is in particular determined by the fact that various approaches and definitions which are equivalent in the finite dimensional situation are, in general, no longer equivalent if the space is infinite dimensional.

In a recent paper Taylor and Tricot [10] introduced packing measures in ℝd. We modify their definition slightly to extend it to a general metric space. Our main concern is to show that in any complete separable metric space every analytic set of non-σ-finite h-packing measure contains disjoint compact subsets each of non-σ-finite measure. The corresponding problem for Hausdorff measures is discussed, but not completely resolved, in Rogers' book [7]. We also show that packing measure cannot be attained by taking the Hausdorff measure with respect to a different increasing function using another metric which generates the same topology. This means that the class of pacing measures is distinct from the class of Hausdorff measures.

The following theorem shows that when packing unit spheres in a large box the spheres occupy at most about 0.7784 of the volume of the box. This improves Rogers' bound [2], which is approximately 0.7796. In the most efficient known packing the ratio is about 0.7405. The box in the theorem could be any bounded solid. Then the (l + 2)(m + 2)(n + 2) becomes the volume of a larger solid all of whose boundary points are at least one unit away from the original solid. At the start of the proof the even larger box could be replaced with a large ball concentric with and radius five units larger than some ball containing the original solid.

Let UK = [0, 1)K be the K-dimensional unit cube, where K ≥ 2. Suppose that we have a distribution ℘ of N points in UK. For × = ( x1, … , xK) ε UK, let A(x) denote the box

and write

Note that since N is the cardinality of ℘ and x1 … xK is the K-dimensional volume of A(x), the term Nx1 … xK represents the “expected number” of, points of ℘ in A(x).

Let

be a real quadratic form in n variables with integral coefficients (i.e., 2fij ε ℤ, fiiε ℤ.) and determinant D ≠ O. A well-known theorem of Cassels [1] states that if the equation f = 0 is properly soluble in integers x1 … , xn then there is a solution satisfying

where F = max |fij and we use the «-notation with an implicit factor depending only on n. More recently it has been shown that f has n linearly independent zeros x1 …, xn satisfying

(see [2, 3 and 6])