Effective Khovanskii, Ehrhart Polytopes, and the Erdős Multiplication Table Problem
Anna Margarethe Limbach, Robert Scheidweiler, Eberhard Triesch
TL;DR
The paper addresses the Erdős Multiplication Table Problem by reformulating $P(k,n)$ via prime-exponent vectors $M_n$ and showing that for each fixed $n$ there exists a polynomial $q_n$ of degree $\\pi(n)$ with $|P(k,n)| = q_n(k)$ once $k$ exceeds an explicit threshold $k_n = n^2 \\cdot \\left(\\prod_{m=1}^{\\pi(n)} \\log_{p_m}(n)\\right) - n + 1$. The authors combine Khovanskii's theorem with Ehrhart theory and apply the effective bound of Granville–Smith–Walker to obtain an explicit leading-coefficient bound $\\mathrm{Vol}(H(M_n)) \\le \\frac{1}{\\pi(n)!} \\prod_{j=1}^{\\pi(n)} \\log_{p_j}(n)$, thereby grounding the polynomiality in a polyhedral-geometry framework. They connect the counting function to lattice polytopes via $Q_n = H(M_n)$ and discuss when $Q_n$ is integrally closed, noting that in general $p(k,n) \\le L(Q_n,k) \\le p(k+\\pi(n),n)$. The work opens avenues for tighter bounds and broader application to other finite subsets of $\\mathbb{Z}^d$, blending additive combinatorics with polyhedral and Ehrhart theory.
Abstract
Let $P(k,n)$ be the set of products of $k$ factors from the set $\{1,\ldots , n\}.$ In 1955, Erdős posed the problem of determining the order of magnitude of $|P (2, n)|$ and proved that $|P (2, n)| = o(n^2 )$ for $n \to\infty$. In 2015, Darda and Hujdurović asked whether, for each fixed $n$, $|P (k, n)|$ is a polynomial in $k$ of degree $π(n)$ - the number of primes not larger than $n$. Recently, Granville, Smith and Walker published an effective version of Khovanskii's Theorem. We apply this new result to show, that for each integer $n$, there is a polynomial $q_n$ of degree $π(n)$ such that $|P (k, n)|=q_n(k)$ for each $k\geq n^2\cdot\left(\prod_{m=1}^{π(n)} \log_{p_m}(n)\right)-n+1.$ Moreover, we give an upper estimate of the leading coefficient of $q_n$.