On some properties of sparse sets: a survey. (English) Zbl 1535.42001
Beliaev, Dmitry (ed.) et al., International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 4. Sections 5–8. Berlin: European Mathematical Society (EMS). 3224-3248 (2023).
Summary: 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.
For the entire collection see [Zbl 1532.00038].
For the entire collection see [Zbl 1532.00038].
MSC:
| 42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |
| 28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 28A80 | Fractals |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 05D10 | Ramsey theory |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
Keywords:
Hausdorff dimension; Fourier dimension; differentiation theorems; growth of Laplace-Beltrami eigenfunctions on manifolds; Euclidean Ramsey theoryReferences:
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