On a problem of Sidon for polynomials over finite fields
Bibliographic record
Abstract
Let $ω$ be a sequence of positive integers. Given a positive integer $n$, we define $$ r_n(ω) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}\colon a,b \in ω, a+b = n, 0 0$ for all $n$ sufficiently large and, for all $ε> 0$, $$ \lim_{n \rightarrow \infty} \frac{r_n(ω)}{n^ε} = 0. $$ P. Erdős proved this conjecture by showing the existence of a sequence $ω$ of positive integers such that $$ \log n \ll r_n(ω) \ll \log n. $$ In this paper, we prove an analogue of this conjecture in $\mathbb{F}_q[T]$, where $\mathbb{F}_q$ is a finite field of $q$ elements. More precisely, let $ω$ be a sequence in $\mathbb{F}_q[T]$. Given a polynomial $h\in\mathbb{F}_q[T]$, we define $$ r_h(ω) = |\{(f,g) \in \mathbb{F}_q[T]\times \mathbb{F}_q[T] : f,g\in ω, f+g =h, \text{deg } f, \text{deg } g \leq \text{deg } h, f\ne g\}|. $$ We show that there exists a sequence $ω$ of polynomials in $\mathbb{F}_q [T]$ such that $$ \text{deg } h \ll r_h(ω) \ll \text{deg } h $$ for $\text{deg } h$ sufficiently large.
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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 | 0.005 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.006 | 0.006 |
| Scholarly communication | 0.004 | 0.016 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.005 | 0.007 |
| Insufficient payload (model declined to judge) | 0.014 | 0.002 |
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".