Effective equidistribution of rational points on horocycle sections in $\text{ASL}(2,\mathbb{Z})\mathrm{\backslash }\text{ASL}(2,\mathbb{R})$

In this paper we prove an effective equidistribution result for both primitive and non-primitive points on certain expanding horocycle sections in ASL$(2,\mathbb{Z})\mathrm{\backslash }$ASL$(2,\mathbb{R})$. This provides an effective version of a result recently proven by the author and generalises recent work by Einsiedler, Luethi and Shah who prove similar results in the context of the unit tangent bundle of the modular surface.