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

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Abstract
Introduction. Let sn 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 In, of Rn. 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, sn greater than n − 1, in Rn. 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 sn > n − 1 is a trivial consequence of combining the result s2 > 1 with the following theorem.

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