Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where
$\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the number of integral points inside (either within or on the boundaries) of $T$ which I denote $N$. A simple computation yields 
$$
N = Area(T) + E
$$ 
with 
$$
|E| \ll |\gamma - \frac{\beta}{\alpha}| + |\beta - \alpha \gamma| 
$$ 
in other words $E$ is bounded by the sum of the two side lengths. Are there ways to get better upper bound than this? Any comments are appreciated.