Note on the integer geometry of bitwise XOR

We consider the set Z0 of non-negative integers endowed with a distance d defined as follows: given x, y ∈ Z0 , d(x, y) is, in binary notation, the result of performing, digit by digit, the “exlusive or” (in short, “XOR”) operation on (the binary notations of) x and y. Dawson, in [?], considers this geometry and suggests the following construction: given k different integers x1, . . . , xk ∈ Z0 , let Vi be the set of integers closer to xi than to any xj with j 6= i, for i, j = 1, . . . , k. Let V = (V1, . . . , Vk) and X = (x1, . . . , xk). V is a partition of {0, 2, . . . , 2n − 1} which, in general, does not determine X. Note that, by definition, V is the Voronoy diagram determined by X. In this paper, we characterize the convex sets of this geometry: they coincide with the line segments and correspond to set intervals through the set of non-zero digits. In particular, the sets of form Vi are convex. Given X and the partition V determined by X, we also characterize in the following terms the ordered sets Y = (y1, . . . , yk) that determine the same partition V: ∀ i = 1, 2, . . . , k, ∀j = 1, 2, . . . , k, j 6= i =⇒ d(yi, xj) , d(xi, yj) > d(yi, xi). This, in particular, extends one of the main results of [?].