A lattice point
is
said to be visible from the origin if no other integer lattice point lies on the line segment joining
the origin and
.
It is a well-known result that the proportion of lattice points visible from the origin is
given by
,
where
denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika
generalized the notion of lattice-point visibility by saying that for a fixed
a lattice
point
is
-visible
from the origin if no other lattice point lies on the graph of a function
, for some
, between the
origin and
.
In their analysis they establish that for a fixed
the proportion
of
-visible lattice
points is
,
which generalizes the result in the classical lattice-point visibility setting. In this paper we give
an
-dimensional
notion of
-visibility
that recovers the one presented by Goins et. al. in two dimensions, and the classical notion in
dimensions. We prove
that for a fixed
the
proportion of
-visible
lattice points is given by
.
Moreover, we give a new notion of
-visibility
for vectors
with nonzero rational entries. In this case, our main result establishes that the proportion of
-visible points
is
, where
is the set of
the indices
for which
.
This result recovers a main theorem of Harris and Omar for
in two dimensions, while showing that the proportion of
-visible
points (in such cases) only depends on the negative entries
of .
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