Given real numbers $a$ and $b$, and an integer $n \geq 2$, let $f(n,a,b)$ be the minimum of $(nint(ja+b)-nint(ia+b)+1)/(j-i)$ (for $1 \leq i < j \leq n$) minus the maximum of $(nint(ja+b)-nint(ia+b)-1)/(j-i)$ (for $1 \leq i < j \leq n$), where $nint$ is the nearest integer function. What is known about the rate at which $f(n,a,b)$ goes to 0 as $n$ goes to infinity, for "generic" real numbers $a, b$? (That is, real numbers whose continued fractions have convergents that grow as dictated by Khinchin's law.) Experiments suggest that $f(n,a,b) = O(1/n^2)$.