Geodesic
On the definition of $B$-points
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 219-236 
P. A. Mozolyako. On the definition of $B$-points. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 219-236. http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a9/
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     title = {On the definition of $B$-points},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a9/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

This paper is devoted to the study of the so-called Bourgain points ($B$-points) of functions in $L^\infty(\mathbb R)$. In 1993, Bourgain showed that for real-valued bounded function $f$ the set $E_f$ of $B$-points is everywhere dense and has maximal Hausdorff dimension, $\dim_H(E_f)=1$; also the vertical variation of the harmonic extension of $f$ to the upper half-plane is finite at $B$-points. An essentially simpler definition of $B$-points is given compared with the original works by Bourgain. A geometric characterization of the $B$-points of Cantor-like sets is obtained. Bibl. – 7 titles.

[1] J. Bourgain, “On the radial variation of bounded analytic functions on the disc”, Duke Math. J., 69:3 (1993), 671–682 | DOI | MR | Zbl

[2] Zh. Burgein, “Ogranichennost variatsii svërtok mer”, Matem. zametki, 54:4 (1993), 24–33 | MR | Zbl

[3] W. Rudin, “The radial variation of analytic functions”, Duke Math. J., 22 (1955), 235–242 | DOI | MR | Zbl

[4] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[5] M. D. O'Neill, “Vertical variation of harmonic functions in upper half spaces”, Colloq. Math., 87 (2001), 1–12 | DOI | MR

[6] P. Jones, “A complete bounded submanifold of $\mathbb C^3$”, Proc. Amer. Math. Soc., 76 (1979), 305–306 | DOI | MR | Zbl

[7] P. A. Mozolyako, V. P. Khavin, O silnoi skhodimosti approksimativnykh edinits, Gotovitsya k publikatsii | Zbl