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

TL;DR

This paper improves bounds on sequenceability in cyclic groups, showing larger subsets can be ordered with distinct partial sums and extending results to the weak sequenceability setting using probabilistic methods.

Contribution

It refines existing bounds for sequenceability in cyclic groups and introduces a new probabilistic approach for the weak sequenceability case.

Findings

Improved the exponent from 1/4 to 1/3 in subset size bounds for sequenceability.

Extended the probabilistic method to establish weak sequenceability with local constraints.

Demonstrated the existence of t-weak sequencings under new bounds using Lovász Local Lemma.

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 Lov\'asz Local Lemma, we obtain the existence of a $t$-weak sequencing whenever $$ t \le \exp\!\big(c(\log p)^{1/4}\big). $$