AI 中文总结
该研究通过构造有限关联结构的初等估计，证明乌曼斯与王提出的等差数列除数覆盖猜想在α<1/2且β<3/8时不成立，尤其在(1/3,1/3)点仍不成立。
AI 中文摘要
n元除数集是指包含1到n中每个整数倍数的有限正整数集合。乌曼斯与王提出了他们强(α,β)除数猜想的等差数列版本：存在项数至多为n^(2β)、大小至多为exp(n^α)的n元除数等差数列。我们无条件证明：高度为H且满足log H = o(√n)的n元除数等差数列，其长度L≥(√(8/27)-o(1))·n^(3/4)/√(log n)。由此，当α<1/2且β<3/8时，该等差数列版本不成立；尤其在提议的(α,β)=(1/3,1/3)点，即便指数级的两个界均放宽n^(o(1))，该猜想仍不成立。证明采用√n以下固定带内的素数，将半素数整除性转化为有限关联结构，再通过初等有界度线性空间估计得出结果。该定理仅涉及一维等差数列版本，未否定高阶强除数猜想。
英文摘要
An \emph{$n$-divisor set} is a finite set of positive integers containing a multiple of every integer from $1$ through $n$. Umans and Wang proposed, as the arithmetic-progression version of their Strong $(α,β)$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2β}$ terms, each of magnitude at most $\exp(n^α)$. We prove unconditionally that an $n$-divisor arithmetic progression of height $H$ with $\log H=o(\sqrt n)$ must have length \[ L\ge \left(\sqrt{\frac{8}{27}}-o(1)\right) \frac{n^{3/4}}{\sqrt{\log n}}. \] Consequently, the arithmetic-progression version is false whenever $α<1/2$ and $β<3/8$. In particular, it is false at the proposed point $(α,β)=(1/3,1/3)$, even if both bounds are relaxed by $n^{o(1)}$ at the exponent level. The proof uses primes in a fixed band below $\sqrt n$ to turn semiprime divisibility into a finite incidence structure. An elementary bounded-degree linear-space estimate then gives the result. This theorem concerns the one-dimensional arithmetic-progression version only; it does not disprove the higher-rank Strong Divisor Conjecture.
