The supremum of autoconvolutions, with applications to additive number theory
Bibliographic record
Abstract
We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if $f\in L^1({\mathbb R}) \cap L^2({\mathbb R})$ is nonnegative and supported on an interval of length $I$, then the supremum of $f\ast f$ is at least $0.631 \|f\|_1^2/I$. This improves the previous bound of $0.591389 \|f\|_1^2/I$. Consequently, we improve the known bounds on several related number-theoretic problems. For a set $A\subseteq\{1,2,\dots,n\}$, let $g$ be the maximum multiplicity of any element of the multiset $\{a_1+a_2 : a_i\in A\}$. Our main corollary is the inequality $g n > 0.631 |A|^2$, which holds uniformly for all $g$, $n$, and $A$.
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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 | 0.003 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".