$\mathbb{Z}_k$中序列性的新界
New bounds for (weak) sequenceability in $\mathbb{Z}_k$
AI总结：
本文改进了循环群$\mathbb{Z}_k$中序列性猜想的界，通过矩形化和单步概率方法，将指数从$1/4$提升至$1/3$，并推广到$t$-弱设置。
摘要：
格雷厄姆的一个著名猜想指出，任何集合$A \subseteq \mathbb{Z}_p \setminus \{0\}$都可以按顺序排列，使得所有部分和都不同。尽管Pham和Sauermann最近证明了对于足够大的素数，该猜想成立，但在一般的阿贝尔群中，甚至在循环群$\mathbb{Z}_k$中仍不成立。对于循环群，已知的最佳结果由Bedert和Kravitz在[4]中给出，他们通过矩形化和两步概率方法证明，对于任何子集$A \subseteq \mathbb{Z}_k \setminus \{0\}$，只要$$ |A| \le \exp\!\big(c(\log p)^{1/4}\big), $$其中$p$是$k$的最小素因子，该猜想成立。在本文中，我们再次使用矩形化方法，随后采用单步概率方法，证明当$$|A| \le \exp\!\big(c(\log p)^{1/3}\big), $$时该猜想成立，从而将指数$1/4$从[4]中改进。此外，同样的单步方法可以适应$t$-弱设置：通过同时施加所有局部约束并应用洛vasz局部引理，我们得到当$$ t \le \exp\!\big(c(\log p)^{1/4}\big) $$时存在$t$-弱序列。
英文摘要：
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 $$ t \le \exp\!\big(c(\log p)^{1/4}\big). $$
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