Exact convex geometry of the residue classes
This site collects work on the convex geometry of multiplication tables modulo N. For a modulus N and a residue a, the lattice points (x, y) in {1, …, N−1}² with xy ≡ a (mod N) form a finite planar set. The areas, boundaries, and vertices of the convex hulls of these sets turn out to be governed exactly by the divisors of a and of N−a. These notes make that structure precise and compute it exactly.
The arguments are elementary, and every construction is checked directly against brute-force computation. The first note is available below; a second is in preparation.
Exact Zero-Class Geometry and Boundary Growth in Multiplication Tables Modulo N
This note studies multiplication tables modulo N through the convex hulls of their residue classes. It introduces the residue-wise area invariant and analyzes two exact structures inside the table. In the zero class, divisor rectangles give a complete convex-hull description, a divisor-chain boundary, and an exact area formula. At the first boundary — the outer frame of the table — a residue-wise four-point geometry gives an explicit lower model for the total area. Together these place exact zero-divisor geometry and cubic-order area growth in a single convex-hull framework.
Zero-Class and Boundary Growth (PDF)A second note, on the convex hulls of modular hyperbolas, gives an exact finite construction of the hull vertices for every nonzero residue, with no coprimality assumption, together with a vertex-count identity that recovers and localizes the classical divisor bound. It will be made available here once it has been publicly posted.