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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 (including 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.

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.

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 (including 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.

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 (including 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.