Zeros of systems of exponential sums and trigonometric polynomials
Bibliographic record
Abstract
Given a subanalytic subset V˜ of the real torus [special characters omitted] and a dense multidimensional trajectory on [special characters omitted], we prove the existence of the mean value of a bounded subanalytic function on V˜ over the isolated intersections of the set V˜ and the trajectory, and give an integral representation for the mean value. We apply this result to obtain an integral representation for the mean value of a quasiperiodic trigonometric polynomial T( x), x ∈ [special characters omitted], over the isolated points of a semitrigonometric set V ⊂ [special characters omitted] (a set described by quasiperiodic trigonometric polynomials). From this integral representation, we conclude that the mean value is equal to zero in the case when the frequencies of T(x) are not compatible with the frequencies of the quasiperiodic trigonometric polynomials that describe V. Next we apply our main result to the complex case, that is, to computing the mean value of an exponential sum over the zeros of a system of n exponential sums in [special characters omitted]. From the integral representation we obtain, we deduce a proof of a generalization of the Gelfond-Khovanskii formula [8] to the case of exponential sums with real frequencies when the frequencies of the sum are not compatible with the frequencies of the system. In the one-dimensional case we give a complete proof of the generalization of the Gelfond-Khovanskii formula to exponential sums with real frequencies. Our arguments represent a combination of real analytic geometry and ergodic theory. They are based on two theorems from completely different parts of mathematics: the cell-decomposition theorem for subanalytic sets, and Weyl's equidistribution law for multidimensional trajectories on the real torus. Weyl's equidistribution law for one-dimensional trajectories is a classical theorem which was published in 1916 [19]. The corresponding law for multidimensional trajectories seems to belong to mathematical folklore. Since we were unable to find a reference, we provide a proof of this theorem, which is a direct generalization of Weyl's original argument.
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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 | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".