Refuting a Conjecture of Umans and Wang on Arithmetic-Progression Divisor Covers

Abstract

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 $(\alpha ,\beta )$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2\beta }$ terms, each of magnitude at most $\exp (n^{\alpha })$. 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 $\alpha <1/2$ and $\beta <3/8$. In particular, it is false at the proposed point $(\alpha ,\beta )=(1/3,1/3)$, 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.