Some computations on the spectra of Pisot and Salem numbers
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Bibliographic record
Abstract
Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdős, Joó and Komornik in 1990, is the determination of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">l(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for Pisot numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l left-parenthesis q right-parenthesis equals inf left-parenthesis StartAbsoluteValue y EndAbsoluteValue colon y equals epsilon 0 plus epsilon 1 q Superscript 1 Baseline plus midline-horizontal-ellipsis plus epsilon Subscript n Baseline q Superscript n Baseline comma epsilon Subscript i Baseline element-of StartSet plus-or-minus 1 comma 0 EndSet comma y not-equals 0 right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi> ϵ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> ϵ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> ϵ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> ϵ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo> ± </mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">l(q) = \inf (|y|: y = \epsilon _0 + \epsilon _1 q^1 + \cdots + \epsilon _n q^n, \epsilon _i \in \{\pm 1, 0\}, y \neq 0).</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> Although the quantity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">l(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is known for some Pisot numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there has been no general method for computing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">l(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This paper gives such an algorithm. With this algorithm, some properties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">l(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its generalizations are investigated. A related question concerns the analogy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">l(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , denoted <inline-formula content-type="math/mathml">
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it