The maximal angular gap among rectangular grid points

Let A - {a₁,...,a_m} and β -{b₁,...,b_n} be two sots of real numbers. Consider the (at, most) mn rays from the origin to the points (a_ib_j), and define the aperture Ap(A, B) to bo the largest angular gap between consecutive rays. Clearly, Ap(A, B) ≥ 2π/mn. Let f(m, n) denote the minimum aperture of any m × n rectangular array, as defined above. In this paper, we show, that for sufficiently large n, f(n, n) < 220/n², so that f(n,n) = Ω(n^-2). We also show that f(m,n) = 2π/mnonly when m = 2, or n = 2 or (m, n) = (4, 4), (4, G) or (6,4).