M ar 1 99 9 Some remarks on the visible points of a lattice

We comment on the set of visible points of a lattice and its Fourier transform, thus continuing and generalizing previous work by Schroeder [1] and Mosseri [2]. A closed formula in terms of Dirichlet series is obtained for the Bragg part of the Fourier transform. We compare this calculation with the outcome of an optical Fourier transform of the visible points of the 2D square lattice. Recently, Mosseri has given a nice and elegant description of the set of visible points of a lattice (i.e., those points except the origin that connect to the origin via a straight line without hitting any other lattice point in between) and some of its properties [2], continuing previous work by Schroeder [1, 3]. Both authors are most interested in the Fourier transform of this set. Unfortunately, the results of these two attempts are contradictory. The algebraic approach of [2] is essentially correct, but contains a couple of mistakes and unnecessary restrictions. In what follows, we add several remarks to straighten this out and then compare the 2D case with an optical experiment. Let Λ = Zb1 ⊕ · · · ⊕Zbn be a lattice in nD space with linearly independent basis vectors b1, . . . , bn. The set FΛ of visible points [4, 3, 2] can be characterized as FΛ = {m1b1 + · · ·+mnbn | gcd(m1, . . . , mn) = 1}, (1) where gcd denotes the greatest common divisor. This set does not include the origin with respect to which it is defined. We follow [2] for the notation as far as possible. The set FΛ is non-periodic and is left invariant by the group of lattice automorphisms, Aut(Λ), which is isomorphic to Gl(n,Z), the group of integer n×n matrices with determinant ±1. This is seen from M ∈ Aut(Λ) transforming fundamental cells of the lattice to other fundamental cells and hence visible points to visible points [4, 6]. As a consequence, the set of visible points admits precisely the same point symmetry as the lattice Λ itself, the transformations are not restricted to pure rotations. This can clearly be seen from Fig. 1 which shows the case of the square lattice. 1 How frequent are visible points? If p denotes the probability of a lattice point to be visible (defined through a volume limit which exists, compare chapter 3.8 of [7]), we also have (l ∈ N = {1, 2, 3, . . .}) the probabilities P (x ∈ l·FΛ) = p ln , (2) because l·FΛ = {m1b1 + · · ·+mnbn | gcd(m1, . . . , mn) = l}. On the other hand, we have Λ = {0} ∪ ∞ ⋃