Simultaneous rational approximation to successive powers of a real number
Bibliographic record
Abstract
We develop new tools leading, for each integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\ge 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , to a significantly improved upper bound for the uniform exponent of rational approximation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove lamda With caret Subscript n Baseline left-parenthesis xi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi> λ </mml:mi> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ξ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat {\lambda }_n(\xi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to successive powers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 comma xi comma ellipsis comma xi Superscript n"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi> ξ </mml:mi> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mi> ξ </mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">1,\xi ,\dots ,\xi ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a given real transcendental number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi"> <mml:semantics> <mml:mi> ξ </mml:mi> <mml:annotation encoding="application/x-tex">\xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As an application, we obtain a refined lower bound for the exponent of approximation to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi"> <mml:semantics> <mml:mi> ξ </mml:mi> <mml:annotation encoding="application/x-tex">\xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by algebraic integers of degree at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n plus 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The new lower bound is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n slash 2 plus a StartRoot n EndRoot plus 4 slash 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n/2+a\sqrt {n}+4/3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a equals left-parenthesis 1 minus log left-parenthesis 2 right-parenthesis right-parenthesis slash 2 asymptotically-equals 0.153"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo> ≃ </mml:mo> <mml:mn>0.153</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a=(1-\log (2))/2\simeq 0.153</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , instead of the current <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n slash 2 plus script upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n/2+\mathcal {O}(1)</mml:annotation>
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How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 teacher head, 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".