Let m, n ≥ 2, m ≤ n. It is well-known that the number of (two-dimensional) threshold functions on an m × n rectangular grid is t(m, n) = 6 π 2 (mn) 2 + O(m 2 n log n) + O(mn 2 log log n) = 6 π 2 (mn) 2 + O(mn 2 log m). We improve the error term by showing that t(m, n) = 6 π 2 (mn) 2 + O(mn 2).
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.
We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in (Z í µí± , +) and (Q + , ⋅) and in certain other groups. Our approach provides a justification for the use of the symbol ⊥ denoting relative primeness in number theory and extends the domain of this convention to some… (More)