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Small-index subsequences in finite cyclic groups

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Chao Liu ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

For r≥1r\ge1, let Ir(n)I_r(n) be the least length forcing a nonempty zero-sum subsequence of index at most rr over the cyclic group CnC_n. We prove Ir(n)≤n1+o(1)I_r(n)\le n^{1+o(1)} when r≥C[ω(n)log⁡(2ω(n))]2r\ge C[\omega(n)\log(2\omega(n))]^2, and I1(pk)≪(k+1)pklog⁡(2pk)I_1(p^k)\ll(k+1)p^k\log(2p^k) for prime powers. We also construct sequences showing that Ir(n)/nI_r(n)/n is unbounded for every fixed rr; for r=1r=1, all terms may be generators.

Provenance statement

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Version history

  1. v1Submitted by Chao LiuInitial depositCurrentOct 30, 2026