Authors:Eric Li (Trinity College, University of Cambridge)
Abstract: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 $\varepsilon>0$, all but $o(x)$ integers $n\le x$ satisfy \[ g_k(n)\ge \left(\frac{3(k-1)}{\log 12}-\varepsilon\right)\log n. \] We also prove, as $n\to\infty$, the pointwise upper bound \[ g_k(n)\le (k-1)\log_2 n+\log_2\log n+O_k(1). \] 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.
From: Eric Li [view email]
[v1]
Mon, 22 Jun 2026 17:47:25 UTC (29 KB)
[v2]
Tue, 23 Jun 2026 12:57:27 UTC (29 KB)
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