Dedekind Sums, the Building Blocks of Lattice-Point Enumeration

Abstract

We encountered Dedekind sums in our study of finite Fourier analysis in Chapter 7, and we became intimately acquainted with their siblings in our study of the coin-exchange problem in Chapter 1 They have one shortcoming, however (which we shall remove): the definition of s(a, b) requires us to sum over b terms, which is rather slow when b = 2100, for example. Luckily, there is a magical reciprocity law for the Dedekind sum s(a, b) that allows us to compute it in roughly \(\log _{2}(b) = 100\) steps in this example. This is the kind of magic that saves the day when we try to enumerate lattice points in integral polytopes of dimension d ≤ 4. In this chapter, we focus on the computational-complexity issues that arise when we try to compute Dedekind sums explicitly. In many ways, the Dedekind sums extend the notion of the greatest common divisor of two integers.