A lattice point of a point lattice 
is a point belonging to 
. If 
is the standard integer
lattice, its lattice points are precisely the points 
whose coordinates 
are all integers.
Thus the definition does not depend on grid lines drawn through the points,
and it applies equally in dimensions greater than two.
The Gauss sphere problem concerns the error in counting lattice points inside a three-dimensional ball.
Wang and Xue (2026, Theorem 1.1) report the sharp upper bound
 |
for a compact full-dimensional convex body 
and a full-rank point lattice 
. The 
are the successive
minima of 
, and 
is the volume of a parallelepiped
spanned by a lattice basis. The reported equality
cases are 
, where 
, 
is a positive integer,
and 
form a basis of 
; thus the defining simplex
has a lattice vertex.
