Assouad-like dimensions of a class of random Moran measures. II. Non-homogeneous Moran sets
Bibliographic record
Abstract
In this paper, we determine the almost sure values of the \Phi -dimensions of random measures \mu supported on random Moran sets in {\mathbb{R}}^d that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets in Hare and Mendivil (2022) to the case of unequal scaling factors. The \Phi -dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the \theta -Assouad spectrum being special cases. The almost sure value of \dim_\Phi \mu exhibits a threshold phenomenon, with one value for “large” \Phi (with the quasi-Assouad dimension as an example of a “large” dimension) and another for “small” \Phi (with the Assouad dimension as an example of a “small” dimension). We give many applications, including both where the scaling factors are fixed and the probabilities are uniformly distributed, and also where the probabilities are fixed and the scaling factors are uniformly distributed.
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How this classification was reachedexpand
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 | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".