  Bounded holomorphic functions on the disk have radial limits in almost every
direction, as follows from Fatou's theorem. Given a zero-measure set $E$ in the
torus $\mathbb T$, we study the set of functions such that $\lim_{r \to 1^{-}}
f(r \, w)$ fails to exist for every $w\in E$ (such functions were first
constructed by Lusin). We show that the set of Lusin-type functions, for a
fixed zero-measure set $E$, contain algebras of algebraic dimension
$\mathfrak{c}$ (except for the zero function). When the set $E$ is countable,
we show also in the several-variable case that the Lusin-type functions contain
infinite dimensional Banach spaces and, moreover, contain plenty of
$\mathfrak{c}$-dimensional algebras. We also address the question for functions
on infinitely many variables.
