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 Z d \mathbb {Z}^{d} ) in the extended d d -dimensional cube n C d = [ − n , + n ] d nC_{d}=\left [ -n,+n \right ] ^{d} on intersecting hyperplanes. In the special case of the hyperplanes { x ∈ R d ∣ x 1 + ⋯ + x d = b } \{ x\in \mathbb {R}^{d}\mid x_{1}+\cdots +x_{d} =b\} , b ∈ Z 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 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 d tends to infinity.