THE MAXIMAL ANGULAR GAP AMONG RECTANGULAR GRID POINTS

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Abstract

Let A = {a1,...,am and B ={b1,...bnbe two sets of real numbers. Consider the (at most) mn rays from the origin to the points (ai,bj)and define the aperture Ap(A, B) to the largest angular gap between consecutive rays. Clearly,Ap(A,B) \geqslant \tfrac{{2\pi }} {{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,Let f(n,n) < \tfrac{{220}} {{n^2 }}, so that f(n, n) =Ω(n^-2). We also show that > f(m,n) =\tfrac{{2\pi }} {{mn}} only when m = 2, or n = 2 or (m, n) = (4,4), (4,6) or (6,4).

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