Direct and inverse problems for restricted signed sumsets in integers
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Bibliographic record
Abstract
Let $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$ $(\leq k)$, we let $h^{\wedge}_{\pm}A = \{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \in \{-1,0,1\} \text{ for } i=0, 1, \ldots, k-1,~~\Sigma_{i=0}^{k-1} |\lambda_{i}|=h\},$ be the $h$-fold restricted signed sumset of $A$. The direct problem for the restricted signed sumset is to find the minimum number of elements in $h^{\wedge}_{\pm}A$ in terms of $\lvert A\rvert$, where $\lvert A\rvert$ is the cardinality of $A$. The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set $A$ for which the minimum value of $|h^{\wedge}_{\pm}A|$ is achieved. In this article, we solve some cases of both direct and inverse problems for $h^{\wedge}_{\pm}A$ in the group of integers. In this connection, we also mention some conjectures in the remaining cases.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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