Bibliographic record
Abstract
The Erdős discrepancy problem asks, “Does there exist an infinite sequence (ti ) i⩾1 taking values in {−1, 1} and a constant c such that |∑1⩽i⩽n tid |⩽c for all ?” Erdős conjectured in the 1930s that no such sequence exists. We examine some variations of this problem with fixed values for c in which the values of d are restricted to particular subsets of . When the values of d are restricted to powers of 2, we show that there are exactly two infinite sequences with discrepancy bounded by 1 and an uncountable number of infinite sequences with discrepancy bounded by 2. When the values of d are restricted to the powers of b for b>2, we show that there is an uncountable number of infinite sequences with discrepancy bounded by 1. We also give a recurrence for the number of sequences of length n with discrepancy bounded by 1. When the values of d are restricted to the odd numbers, we conjecture that there are exactly four infinite sequences with discrepancy bounded by 1 and give some experimental evidence for this conjecture.
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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 | 0.004 | 0.044 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.011 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".