A note on the Besicovitch dimension of the closest packing of spheres in R_n

Extract

Introduction. Let s_n denote the infimum of the set of real numbers X, where x belongs to X if, and only if, x is the Besicovitch dimension of the residual set of a packing of n-spheres into the unit n-cube I_n, of R_n. In recent work ((1)) I have shown that s₂ is greater than one, and, quite naturally, I have since been asked whether or not the proof can be generalized to prove the analogous result, s_n greater than n − 1, in R_n. Whilst it is clear, in theory, that this could be done, in practice the details might become rather complicated. However, such a generalization is unnecessary, for the result s_n > n − 1 is a trivial consequence of combining the result s₂ > 1 with the following theorem.