Local dimensions of measures of finite type II - Measures without full\n support and with non-regular probabilities
Bibliographic record
Abstract
Consider a sequence of linear contractions $S_{j}(x)=\\varrho x+d_{j}$ and\nprobabilities $p_{j}>0$ with $\\sum p_{j}=1$. We are interested in the\nself-similar measure $\\mu =\\sum p_{j}\\mu \\circ S_{j}^{-1}$, of finite type. In\nthis paper we study the multi-fractal analysis of such measures, extending the\ntheory to measures arising from non-regular probabilities and whose support is\nnot necessarily an interval.\n Under some mild technical assumptions, we prove that there exists a subset of\nsupp$\\mu $ of full $\\mu $ and Hausdorff measure, called the truly essential\nclass, for which the set of (upper or lower) local dimensions is a closed\ninterval. Within the truly essential class we show that there exists a point\nwith local dimension exactly equal to the dimension of the support. We give an\nexample where the set of local dimensions is a two element set, with all the\nelements of the truly essential class giving the same local dimension. We give\ngeneral criteria for these measures to be absolutely continuous with respect to\nthe associated Hausdorff measure of their support and we show that the\ndimension of the support can be computed using only information about the\nessential class.\n To conclude, we present a detailed study of three examples. First, we show\nthat the set of local dimensions of the biased Bernoulli convolution with\ncontraction ratio the inverse of a simple Pisot number always admits an\nisolated point. We give a precise description of the essential class of a\ngeneralized Cantor set of finite type. Lastly, we study a maximal loop class\nthat is not truly essential.\n
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".