Smooth Billiards with a Large Weyl Remainder
Bibliographic record
Abstract
We prove an estimate from below for the remainder in Weyl's law for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. The proof is based on the analysis of the singularity in the wave trace, occurring at time |$T>0$| equal to the length of the closed trajectories belonging to the family. Such families of periodic orbits produce large singularities in the wave trace, which in turn imply estimates from below for the absolute value of the Weyl remainder averaged over a spectral window. We also obtain lower bounds for the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".