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

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 $$|A|\le \exp (c(\log p{)}^{1/4}),$$ 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 $$|A|\le \exp (c(\log p{)}^{1/3}),$$ 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 $$t\le \exp (c(\log p{)}^{1/4}).$$