GENERAL FORMS FOR MINIMAL SPECTRAL VALUES FOR A CLASS OF QUADRATIC PISOT NUMBERS
Bibliographic record
Abstract
This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as l1(q), and this definition is extended to all height m polynomials as ≔ɛɛɛɛɛlm(q) ≔ inf(|y|:y=ɛ0+ɛ1q1+…+ɛnqn, ɛi∈Z, |ɛi| ⩽ m, y≠0). A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of lm(q) when q is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows. THEOREM. Let q be a quadratic Pisot number that satisfies a polynomial of the form p(x) = x2−ax ± 1, with conjugate r. Let q have convergents {Ck/Dk} and let k be the maximal integer such that |Dkr−Ck|⩽m11−|r|; then A value related to l(q) is a(q), the minimal non-zero value when all ±1 polynomials are evaluated at q. Formally, this is An open question concerning how often a(q) = l(q) is also answered in this paper.
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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 | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".