I have a hyper-Cube lattice with coordinates between 0 and 10, therefore 11^N lattice points. I take one of this point, C, and then another point P at distance D from C. I need to know a good algorithm or even a close formula that gives me the number of lattice points that have a distance from C equal or less than D. Since N, the dimensionality of the problem, could be of the order of the hundreds, I would prefer not to examine the distance of all 11^N points from D.
What have I done so far? Not much, to be honest. I my mind this problem is equivalent to counting the number of lattice points in the intersection between the hyperspher centered in C passing through P and the hypercubewith side 11 and a corner in the origin.
I tried the low dimensionality approach to see if a pattern emerges but already the 1 dimensional problem has too many "IF" conditions.
Let's say I am interested in a good approximation of the result, then i could calculate the intersection of a N-Sphere and an N-Cube but it would be too imprecise for small values.
Are there some easy formulas for this kind of problem? From the other pages I find in stackechange the problem seems quite hard but maybe the box constraint, instead of being a difficulty, simplifies something