Strichartz’s spectral remainder conjecture for compact Heisenberg manifolds

Let $M=\Gamma\backslash\mathbb{H}_d$ be a compact Heisenberg manifold, let $N_M(\lambda)$ be the spectral counting function of its sub-Laplacian, and write $R_M(\lambda)=N_M(\lambda)-A_d\operatorname{vol}(M)\lambda^{d+1}$ . Strichartz's conjecture asserts that, as $\lambda\to\infty$ , $R_M(\lambda)=O_M(\lambda^d)$ ; equivalently, there exists a constant $C_M>0$ such that $|R_M(\lambda)|\le C_M\lambda^d$ for all sufficiently large $\lambda$ .

References

Primary source

arXiv

Additional references

Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A September 2026 preprint says the predicted error size is too small, but the exact logarithmic correction remains unknown.

Strichartz’s conjecture concerns the remainder in spectral counting for compact Heisenberg manifolds and predicts an error of order $O(\lambda^d)$ . The new result claims this polynomial order is unavoidable but that the conjectured logarithmic factor is not sharp.

September 2026 spectral bounds

A preprint titled Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds claims lower bounds excluding the conjectured $O(\lambda^d)$ remainder and an upper bound improving the logarithmic factor. The sharp logarithmic factor remains open, and the result is not yet peer reviewed.

Current status (as of September 2026): The conjectured $O(\lambda^d)$ remainder is claimed to be false, while the sharp logarithmic remainder remains unresolved and the new bounds await verification.

Sources

No solutions have been posted yet.