On the definition of $B$-points

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.