We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the 2-torus and the modular surface.

中文翻译：

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圆环模曲面产品中扩展horocycles的原始理性点

我们证明了原始有理点和由单项式定义的原始有理点在环面和模曲面的乘积中沿长星环轨道的有效等分布。这回答了第一位和最后一位作者与 Shahar Mozes 和 Uri Shapira 在联合工作中提出的问题。在一定的全等条件下，我们证明了共轭有理点在2-环面和模曲面上的联合等分布。