Inverse Falconer Distance Theorems over the Integer Residue Rings $\mathbb{Z}_n$

TL;DR

This paper proves an inverse theorem for the Falconer distance problem over integer residue rings, revealing that sets with few distances exhibit strong algebraic structure supported on annihilator submodules.

Contribution

It introduces a novel ideal-theoretic rigidity principle and a divisor-depth decomposition for $Z_n$, providing the first inverse theorem for composite moduli.

Findings

Sets with few distances are supported on cosets of annihilator submodules.

Near-extremizers exhibit algebraic degeneracy in the distance form.

Complete classification of near-extremizers shows a rigidity phenomenon.

Abstract

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure supported on annihilator submodules arising from the arithmetic of $n$. As a consequence, we obtain the first inverse theorem for the Falconer distance problem over $\mathbb{Z}_n$ for composite moduli. We show that if a set $E \subset \mathbb{Z}_n^d$ of size $|E| \asymp n^{(d+1)/2}$ determines only $O(n)$ distinct squared distances, then $E$ must be supported on a coset of an annihilator submodule on which the distance form is algebraically degenerate. The proof introduces a divisor-depth decomposition intrinsic to $\mathbb{Z}_n$, together with a lifting mechanism that transfers local degeneracies at prime moduli into global ideal-theoretic constraints. This yields a complete classification of near-extremizers for the Falconer distance problem in the ring setting, revealing a rigidity phenomenon with no analogue over fields.