An old conjecture of Erdos–Turán on additive bases
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Bibliographic record
Abstract
There is a 1941 conjecture of Erdős and Turán on what is now called additive basis that we restate: <bold>Conjecture 0.1</bold> (Erdős and Turán) <bold>.</bold> Suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 equals delta 0 greater-than delta 1 greater-than delta 2 greater-than delta 3 midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:msub> <mml:mi> δ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:msub> <mml:mi> δ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:msub> <mml:mi> δ </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:msub> <mml:mi> δ </mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo> ⋯ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">0 = \delta _0>\delta _1>\delta _2>\delta _3\cdots</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an increasing sequence of integers and <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s left-parenthesis z right-parenthesis colon equals sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts z Superscript delta Super Subscript i Superscript Baseline period"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:munderover> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> δ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">s(z) : = \sum _{i=0}^\infty z^{\delta _i}.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> Suppose that <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s squared left-parenthesis z right-parenthesis colon equals sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts b Subscript i Baseline z Superscript i Baseline period"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:munderover> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">s^2(z) := \sum _{i=0}^\infty b_i z^i.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Subscript i Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b_i>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i"> <mml:semantics> <mml:mi>i</mml:mi> <mml:annotation encoding="application/x-tex">i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace b Subscript n Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{b_n\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is unbounded. Our main purpose is to show that the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace b Subscript n Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{b_n\}</mml:annotation> </mml:sema
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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 | 0.001 | 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