Self-similar sets and self-similar measures in the $p$-adics
Bibliographic record
Abstract
In this paper, we investigate p -adic self-similar sets and p -adic self-similar measures. We introduce a condition (C) under which p -adic self-similar sets can be shown to have a number of nice properties. It is shown that p -adic self-similar sets satisfying condition (C) are p -adic path set fractals. This allows us to easily compute the Hausdorff dimension of these sets. We further show that the set of p -adic path set fractals is strictly larger than this set of p -adic self-similar sets. The directed graph associated to p -adic self-similar sets satisfying condition (C) is shown to have a unique essential class. Moreover, it is shown that almost all points are eventually in the essential class. For p -adic self-similar measures satisfying this condition, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. We next study the more general p -adic path set fractals, first showing that the existence of an interior point is equivalent to the set having Hausdorff dimension 1 . We further show that often the decimation of p -adic path set fractals results in a set with maximal Hausdorff dimension.
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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 | 0.002 |
| 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 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 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".