We study questions concerning the distribution of lattice points in dimensions two and higher. We give asymptotic formulas for the number of integer lattice points of fixed index visible from certain admissible sets. We also study the shape of the body
P
A
1
,
…
,
P
A
k
PA_1, \dots , PA_k
when the lattice points
A
1
,
…
,
A
k
A_1, \dots , A_k
move inside a given large ball, and
P
P
is a lattice point visible from all
A
i
A_i
’s.