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

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