Shah conjecture on horocycle flow at fractional-power times

Prove that the horocycle flow evaluated at the times \(\{n^{1+\eta}: n\in\mathbb{N}\}\) is equidistributed for some \(\eta>0\), beyond the ranges already established for particular classes of lattices.

Background

The paper discusses sparse polynomial-type sampling of horocycle flows and identifies a conjecture attributed to Shah concerning times of the form n1+ηn^{1+\eta}. The question asks whether equidistribution persists when the sampling sequence is substantially sparser than the integers.

The text records progress by Venkatesh for cocompact lattices and later improvements by Flaminio, Forni, and Tanis to 0<η<1/120<\eta<1/12. Thus, the quoted conjecture is not wholly resolved in the generality suggested by the discussion, despite these partial results.

References

While equidistribution of the full set of integer times N of the horocycle flow in quotients of SL2(R) follows from the corresponding continuous statements, much less is known once one shifts to a sparser subset of N. Margulis conjectured that the horocycle flow at the subset of primes should be equidistributed, while Shah conjectured the same should be true for the subset {n1+η | n ∈ N} for some η > 0.

— 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