Sharpness of the two Cauchy–Schwarz bounds

Determine whether the two applications of the Cauchy–Schwarz inequality used in the sharpened geometry-of-numbers estimate are tight for displacement vectors arising from generalized modular embeddings, or establish a sharper bound for those vectors.

Background

The sharpened exponent √2(r−1)(1−δ) results from applying Cauchy–Schwarz twice to the conjugate displacement vectors of two traces. The paper notes that these vectors are not arbitrary but arise from a generalized modular embedding, leaving open whether their additional structure permits a better estimate.

References

Whether the two applications of Cauchy-Schwarz in the proof above are themselves tight for the displacement vectors $(\ell_2,\ldots,\ell_r)$ that actually arise from a generalized modular embedding, or whether a sharper bound exists for these specific vectors, we do not know.

— Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension  (2608.17604 - Zuevsky, 18 Aug 2026) in Remark following Proposition 4.3, Section 4