New bounds for (weak) sequenceability in $\mathbb{Z}_k$

TL;DR

The paper tackles Graham's conjecture for cyclic groups by studying sequenceability and $t$-weak sequenceability in $A\subseteq \mathbb{Z}_k\setminus\{0\}$. It combines a rectification-based Structure Theorem with a unified one-shot probabilistic framework that uses random splittings of dissociated blocks and the Lovász Local Lemma to avoid zero-sum short intervals, achieving sharper bounds. The main contributions are a bound $|A|\le\exp\big(c(\log p)^{1/3}\big)$ for sequenceability and a bound $t\le\exp\big(c(\log p)^{1/4}\big)$ for $t$-weak sequenceability in terms of the least prime divisor $p$ of $k$, along with an improved classical sequencing bound via the same method. This work advances quantitative understanding of Graham-type problems in cyclic groups and introduces a streamlined one-shot method that tightens prior exponents.

Abstract

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and Sauermann in [16], it remains open for general abelian groups, even in the cyclic case $\mathbb{Z}_k$. For cyclic groups, the best known result is due to Bedert and Kravitz in [4], who proved - using a rectification and a two-step probabilistic approach - that the conjecture holds for any subset $A \subseteq \mathbb{Z}_k \setminus \{0\}$ such that $$ |A| \le \exp\!\big(c(\log p)^{1/4}\big), $$ for some constant $c>0$, where $p$ denotes the least prime divisor of $k$. In this paper, we improve their bound using a rectification argument again, followed by a one-shot probabilistic approach, showing that the conjecture holds whenever $$|A| \le \exp\!\big(c(\log p)^{1/3}\big), $$ thus improving the exponent $1/4$ from [4]. Moreover, the same one-shot approach adapts to the $t$-weak setting: by imposing all local constraints at once and applying the Lovász Local Lemma, we obtain the existence of a $t$-weak sequencing whenever $$ t \le \exp\!\big(c(\log p)^{1/4}\big). $$

Theorems & Definitions (14)

• Conjecture 1.1: GR and EG
• Theorem 1.2: BK
• Theorem 1.3: Improved classical bound
• Theorem 1.4
• Lemma 2.1
• Theorem 2.2: Structure Theorem BK
• Proposition 2.3
• Lemma 2.4: Lemma 5.1 of BK
• Lemma 2.5
• proof