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

TL;DR

This paper refines the asymptotic estimate for the number of threshold functions on a two-dimensional rectangular grid, reducing the error term to a tighter bound.

Contribution

It improves the known asymptotic formula by sharpening the error term from logarithmic factors to a simpler bound.

Findings

The number of threshold functions is asymptotically / (mn)^2.

The error term is improved to O(mn^2).

The main term remains / (mn)^2 with a tighter error estimate.

Abstract

Let $m,n\ge 2$, $m\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\times n$ rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})= \frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by showing that $$ t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $$