Prime-Power Rarefaction and a Density-One Lower Bound for Erdős Problem 400

For fixed $k\ge 2$, let $g_k(n)$ be the greatest excess $a_1+\cdots +a_k-n$ among positive integers $a_i$ satisfying $a_1!\cdots a_k!\mid n!$. We prove that, for every $\epsilon >0$, all but $o(x)$ integers $n\le x$ satisfy $$g_k(n)\ge (\frac{3(k-1)}{\log 12}-\epsilon )\log n\mathrm{.}$$ We also prove, as $n\to \mathrm{\infty }$, the pointwise upper bound $$g_k(n)\le (k-1){\log }_{2}n+{\log }_2\log n+O_k(1)\mathrm{.}$$ The central analytic input is uniform phase separation for one or two frequencies on fixed-prime $S$-unit progressions, deduced directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer, and the resulting uniform digit-sum normal-order theorem. A mixed $2$–$3$ representation, quantitative two-block estimates, and a large-prime Kummer sieve produce the stated coefficient.