Margulis conjecture on horocycle flow at prime times

Prove that the horocycle flow on quotients of \(\mathrm{SL}_2(\mathbb{R})\) evaluated at prime times is equidistributed.

Background

The paper situates its density theorem within the broader study of sparse-time evaluations of horospherical and horocycle flows. It explains that equidistribution at all integer times follows from continuous-time results, whereas substantially less is known for sparse sequences such as the primes.

The cited conjecture, attributed to Margulis, concerns whether sampling the horocycle flow at prime times preserves equidistribution. The paper notes that subsequent work has established partial non-concentration and almost-prime results, but the prime-time equidistribution problem itself is presented as a conjectural unresolved question.

References

Margulis conjectured that the horocycle flow at the subset of primes should be equidistributed.

— Density of almost squares of horospherical orbits in non-uniform quotients of $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$  (2609.24752 - Andritsch, 21 Sep 2026) in Section 1, Introduction, page 1