integer points in a face of $n-$ dimensional Box

Question

Let $r,n$ be even positive integers. Consider a box of the form $B_n=[2^{r-1},2^{r}-1]^{n/2}\times [2^{r/2-1},2^{r/2}-1]^{n/2}.$ I want to find a hyperplane with the maximum number of points in $B_n.$ It is enough to search for such hyperplanes in the faces of the box $B_n$ (I think). Here not all faces have the same number of points. I want to compute the maximum number of points that a face has. Any idea? (for instance if $n=2,r=4$, such a face has $8$ points)

Answer 1

Yes, you can just check the faces and in particular you want the one with all the first factor and all but one of the second because the first is larger than the second. If we consider $n=4, r=6$ we have $B_4=[32,63]\times[32,63]\times [4,7] \times [4,7]$. The largest hyperplane will be $[32,63]\times[32,63]\times [4,7]$. There will be many of that size, not all faces of the box, but there won't be any with more points than that.