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Record W4389775472 · doi:10.4171/icm2022/153

On some properties of sparse sets: a survey

2023· book-chapter· en· W4389775472 on OpenAlex
Malabika Pramanik

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.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.
fundA Canadian funder is recorded on the work.

Bibliographic record

VenueEMS Press eBooks · 2023
Typebook-chapter
Languageen
FieldMathematics
TopicMathematical Dynamics and Fractals
Canadian institutionsUniversity of British Columbia
FundersDivision of Mathematical SciencesNatural Sciences and Engineering Research Council of CanadaBanff International Research Station for Mathematical Innovation and DiscoveryAssociation for Women in Mathematics
KeywordsCardinality (data modeling)MathematicsMeasure (data warehouse)Euclidean spaceEuclidean geometrySpace (punctuation)Ergodic theoryDimension (graph theory)Pure mathematicsFractalComputer scienceGeometryMathematical analysis

Abstract

fetched live from OpenAlex

Sparse sets are, by definition, sets that are small, either in cardinality, measure, dimension, or density. Curves, surfaces, and other submanifolds are standard examples of sparse sets in Euclidean space. However, many sparse sets naturally occurring in ergodic and geometric measure theory, such as Cantor-like sets or self-similar fractals, lack the regularity of the aforementioned objects. Despite this deficiency, many sparse sets are rich in arithmetic, geometric, and analytic properties that can be viewed as working substitutes for smoothness. This has led to a vibrant line of inquiry into the governing principles behind certain phenomena that are typically associated with submanifolds and that have the potential for ubiquity in far more general contexts. Structural and analytical properties of sparse sets, whether discrete or continuous, lie at the center of many problems in harmonic analysis, fractal geometry, combinatorics, and number theory. This is a survey of a few such problems that the author has worked on.

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 imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMeta-epidemiology (narrow)
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Other · Consensus signal: Other
Teacher disagreement score0.006
Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0010.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.233
GPT teacher head0.309
Teacher spread0.076 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it