Joint equidistribution on the product of the circle and the unit cotangent bundle of the modular surface

Translates of rational points along expanding closed horocycles on the modular surface

“…The equidistribution for the case was later proved by Einsiedler–Luethi–Shah [ 8 ]; Jana [ 16 , Theorem 1] recently gave an alternative spectral proof to this equidistribution result. We also mention that both [ 5 , Theorem 2] and [ 16 , Theorem 1] are valid in the same setting as [ 8 ], namely, on the product of the unit tangent bundle of the modular surface and a torus. When the equidistribution fails as the aforementioned symmetry implies that is always trapped in the closed horocycle .…”

“…Our proofs of Theorems 1.1 and 1.2 rely on spectral estimates collected in the recent paper of Kelmer and Kontorovich [ 18 ], with a necessary refinement of [ 18 , (3.6)] in the form of Proposition 3.3 , which comes at the cost of a higher degree Sobolev norm. This strategy is standard and is also found in [ 4 , 16 , 27 , 31 ], to name just a few recent papers on related problems. The analysis in [ 18 ] was carried out in a more general setting, namely for the congruence covers with p a prime number.…”