Bilinear spherical maximal function with fractal dilations

In this paper, we investigate $L^p-$boundedness of the bilinear spherical maximal function associated with a general set $E\subset {\mathbb{R}}_{+}$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of upper Minkowski dimension of the set $E$. A particular case of this study, settles an open question of $L^p-$boundedness of the lacunary bilinear spherical maximal function at borderline cases $p_1=1$ or $p_2=1$ in dimension $d\ge 3$.