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Establish limit distributions of zeros for congruence groups and prove weak convergence to predicted measures

Prove, for interesting congruence subgroups Γ, that the normalized counting measure on zeros in even weight k converges weakly as k→∞ to the pushforward of arc-length on the corresponding unit-circle segments under the Möbius maps that generate the Hausdorff-limit geodesic configuration, as specified by the formula proposed in the paper.

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Background

Beyond describing the Hausdorff-limit geodesic set, the authors propose a specific weak limit for the zero-counting measures, built from Haar measure on the unit circle transported by the linear-fractional maps defining the limiting geodesics.

They ask whether this measure-limit statement holds, extending their equidistribution result in angle proved for Γ(N) with odd N.

References

We list some open problems. Find the limit distribution of zeros of Eisenstein series for interesting congruence groups. More specifically, suppose that $ \mathrm{Ls} Z_{k,\Gamma}$ is the union of geodesic segments given as image of segments connected $I_\gamma$ of the unit circle under $\gamma \in \mathrm M_2(\mathbb Z) \cap \mathrm{GL}2(\mathbb Q)$ for $\gamma \in \mathcal I$, $\mathcal I$ a finite set; is it true that the normalized counting measure supported at the zeros in even weight $k$, converges weakly $$\frac{1}{\# \overline Z{k,\Gamma}\sum_{z \in \overline Z_{k,\Gamma} \delta_z \rightarrow \sum_{\gamma \in \mathcal I} \frac{1}{\mu_{S_1}(I_\gamma)} \gamma_* \mu_{S1}|{I\gamma} \ (k \in 2 \mathbb Z_{>1}, k \rightarrow + \infty)$$ where $\mu_{S1}$ is the Haar measure on $S1$?

— Geodesic clustering of zeros of Eisenstein series for congruence groups (2509.16108 - Santana et al., 19 Sep 2025) in Section: Open problems (final section)