Improved Bounds for the Index Conjecture in Zero-Sum Theory

TL;DR

This work advances the Index Conjecture in zero-sum theory for 4-term minimal zero-sum sequences by sharpening the general upper bound to $4.6\times10^{13}$ and showing improved bounds under additional coprimality conditions, while also validating the conjecture computationally for all $n<1.8\times10^{6}$ via high-performance computing. The approach blends Fourier-analytic techniques with carefully bounded exponential sums, using a smoothed indicator $f$ and associated coefficients $\hat{f}$ to lower-bound the relevant sums $S_1$ and upper-bound $S_0$, and optimizing a smoothing parameter $H$ (taken near $7000$). Key theoretical contributions include a structured decomposition of index-related sums, injectivity arguments relating residue-pairs, and explicit numerical bounds that drive the improved thresholds. The HPC verification complements the analytic bounds, illustrating the conjecture’s validity across a broad range and highlighting the practical power of parallel computation in number-theoretic conjectures of zero-sum type.

Abstract

The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for $n>10^{20}$. In this paper, we prove that said upper bound can be reduced to $4.6\cdot10^{13}$, and lower under certain coprimality conditions. Further, we verify the conjecture for $n<1.8\cdot10^6$ through the application of High Performance Computing (HPC).

Theorems & Definitions (21)

• Definition 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof