The number of gridpoints on hyperplane sections of the $d$-dimensional cube

Abstract:

We deduce a formula for the exact number of gridpoints (i.e., elements of $\mathbb {Z}^{d}$) in the extended $d$-dimensional cube $nC_{d}=\left [ -n,+n \right ] ^{d}$ on intersecting hyperplanes. In the special case of the hyperplanes $\{ x\in \mathbb {R}^{d}\mid x_{1}+\cdots +x_{d} =b\}$, $b\in \mathbb {Z}$, these numbers can be written as a finite sum involving products of certain binomial coefficients. Furthermore, we consider the limit as $n$ tends to infinity which can be expressed in terms of Euler-Frobenius numbers. Finally, we state a conjecture on the asymptotic behaviour of this limit as the dimension $d$ tends to infinity.