Primitive rational points on expanding horocycles in products of the modular surface with the torus

Abstract

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 two-torus and the modular surface.

arXiv commentFixed a few typos and improved the exposition of some of the more technical discussions

References in corpus (5)

• Equidistribution of primitive rational points on expanding horospheres
• Distribution of Shapes of orthogonal Lattices
• The value distribution of incomplete Gauss sums
• Effective equidistribution of primitive rational points on expanding horospheres
• Primitive rational points on expanding horospheres in Hilbert modular surfaces

Cited by in corpus (3)

• Translates of rational points along expanding closed horocycles on the modular surface
• Joint equidistribution on the product of the circle and the unit tangent bundle of the modular surface
• Rational points on non-linear horocycles and pigeonhole statistics for the fractional parts of $\sqrt{n}$