Asymptotics for Numbers of Line Segments and Lines in a Square Grid

Abstract

We present an asymptotic formula for the number of line segments connecting q + 1 points of an n × n square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q = 2 is so generalized.

Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

“…We can also drop out the logarithms from ( 4) and (5). Generalizing (6), we will in Section 4 prove that…”

“…The proof of (19) can be extended to show that f q (m, n) = 6 π 2 q 2 (mn) 2 + O(mn 2 ) if m ≤ n. Also the formulas (6), (7) and (8) for f (n) generalize. We have [5] f q (n) = 6 π 2 q 2 n 4 + r(n),…”

“…Besides the number of threshold functions, there are several other quantities whose asymptotics can be studied in a similar way. For example, asymptotic formulas for the number of gridlines [3] and q-gridlines [5] are well-known in G(n). A simple modification of our above procedure gives such formulas in G(m, n).…”

“…In practical computation of all these quantities, recursive formulas [8,9,3,4] are useful. They have also been applied in computer experiments to find asymptotic formulas.…”

Asymptotics of the number of 2-threshold functions

“…Configurations of planar lattice points inside a circle are also considered in [15,16] and configurations of lattice points inside a sphere are considered in [24]. Configurations produced by straight lines are also intensively studied [5,18,12,2,13,20]. This is partially because of their applications in digital geometry and computer graphics.…”