Local dimensions of measures of finite type on the torus
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
The structure of the set of local dimensions of a self-similar measure has been studied by numerous mathematicians, initially for measures that satisfy the open set condition and, more recently, for measures on $\mathbb{R}$ that are of finite type. In this paper, our focus is on finite type measures defined on the torus, the quotient space $\mathbb{R}\backslash \mathbb{Z}$. We give criteria which ensures that the set of local dimensions of the measure taken over points in special classes generates an interval. We construct a non-trivial example of a measure on the torus that admits an isolated point in its set of local dimensions. We prove that the set of local dimensions for a finite type measure that is the quotient of a self-similar measure satisfying the strict separation condition is an interval. We show that sufficiently many convolutions of Cantor-like measures on the torus never admit an isolated point in their set of local dimensions, in stark contrast to such measures on $\mathbb{R}$. Further, we give a family of Cantor-like measures on the torus where the set of local dimensions is a strict subset of the set of local dimensions, excluding the isolated point, of the corresponding measures on $\mathbb{R}$.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it