  This work aims to extend the existing results on the Hausdorff dimension of
the classical thick point sets of a Gaussian free field (GFF) to a more general
class of exceptional sets. We adopt the circle or sphere averaging
regularization to treat a singular GFF in any dimension, and introduce the
notion of "$f-$steep point" of the GFF for certain test function $f$. Roughly
speaking, the $f-$steep points of a generic element of the GFF are locations
where, when weighted by the function $f$, the "steepness", or in other words,
the "rate of change" of the regularized field element becomes unusually large.
Different choices of $f$ lead to the study of various exceptional behaviors of
the GFF. We investigate the Hausdorff dimension of the set consisting of
$f-$steep points, from which we can recover the existing results on thick point
sets for both log-correlated and polynomial-correlated GFFs, and also obtain
new results on exceptional sets that, to our best knowledge, have not been
previously studied. Our method is inspired by the one used to study the thick
point sets of the classical 2D log-correlated GFF.
