The multiplication table problem in large dimensions

Abstract: For $N\geq 2$ and $k\geq 1$, let $M_k(N):=#{x_1\cdots x_k : x_i\in{1,\ldots,N}\text{ for all } i}$ be the $k$-dimensional multiplication table. Given $N$, Khovanskii's theorem implies that $M_k(N)$ agrees, for all sufficiently large $k$, with a polynomial in $k$ of degree $π(N)$. We determine the asymptotic size of its leading coefficient, proving that, as $N\to\infty$, with $k$ sufficiently large relative to $N$, [ M_k(N) = \exp\bigg((2π+o(1))\frac{\sqrt{N}}{\log N}\bigg)\frac{k^{π(N)}}{π(N)!}. ] We also study the analogous problem when the factors are restricted to $y$-smooth integers. For $y=o(\log N)$, we prove that the number of distinct products of $k$ such integers up to $N$ is asymptotic to the number of $y$-smooth integers up to $N<sup>k$, uniformly for $k\geq 1$.

• The paper determines an asymptotic formula for the leading coefficient of the polynomial describing the number of distinct products in the k-dimensional multiplication table problem as the dimension k grows with N, showing that log V_N ~ 2π√N/ log N.
• A uniform-in-k asymptotic result for M_k(N, y) is shown, using Erdős's problem's connection with Ehrhart-type theory of lattice polytopes and providing new insights into the distribution of primes.
• The authors present explicit geometric and algebraic methods for bounding and evaluating the volume of the convex hull of multiplication table entries, giving the leading exponential scale for the normalized volume of this region.

Overview and main results

This paper by Sabuncu and Tâfula studies the $k$-dimensional multiplication table problem in the regime where the dimension $k$ grows with $N$ (2608.16163). For $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$, Khovanskii's theorem on iterated sumsets implies that for each fixed $N$ there exists $k_N$ such that $M_k(N)=\mathcal P_N(k)$ for all $k\ge k_N$, where $\mathcal P_N$ is a polynomial of degree $\pi(N)$ with leading coefficient equal to the normalized volume $k$0 of the convex hull of exponent vectors of integers up to $k$1. The paper's central result determines the asymptotic size of this coefficient:

$k$2

so that, uniformly once $k$3 is sufficiently large relative to $k$4, $k$5 This is a sharp contrast with the fixed-$k$6 results of Koukoulopoulos, which give order-of-magnitude estimates only; here the polynomial structure forced by Khovanskii's theorem permits an exact asymptotic for the exponential scale.

The naive bounds from the ambient simplex $k$7 and its inscribed lattice simplex give $k$8 between constants times $k$9 and $N$0. The theorem asserts the correct order is the smaller one — a nontrivial fact, since the trivial upper bound overestimates by a factor of roughly $N$1 in the exponent.

The paper also treats a smooth variant: for $N$2 and uniformly in $N$3, $N$4 (Theorem 1). This extends Mehdizadeh's $N$5 result, valid up to $N$6, to all dimensions but a shorter smoothness range; the authors explicitly pose as an open problem how far $N$7 can be enlarged while the uniform-in-$N$8 asymptotic persists.

Method for the smooth variant

The proof of Theorem 1 rests on two observations. First, the leading coefficient satisfies $N$9 throughout $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$0: since $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$1 uniformly over $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$2, the floors in the inscribed simplex contribute only $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$3. Second, every product counted by $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$4 can be factored into $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$5 factors each at most $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$6, giving

$M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$7

and standard smooth-number asymptotics (Ennola's theorem for $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$8; the de la Bretèche–Tenenbaum framework for larger $M_k(N):=\#\{x_1\cdots x_k : x_i\in[N]\}$9) show the two extreme quantities are $N$0-equivalent uniformly in $N$1. In this range the convex hull is thus asymptotically indistinguishable from the weighted simplex $N$2, and distinctness of products imposes no loss beyond the size restriction $N$3.

Volume bounds via geometric decomposition

For the main theorem, the key structural input is the decomposition

$N$4

which exploits the fact that any $N$5 has at most one prime factor exceeding $N$6. Grouping large primes into dyadic-type blocks $N$7 between $N$8 and $N$9, the authors construct explicit lower and upper model bodies $k_N$0 and an upper comparison body $k_N$1 built from scaled coordinate simplices $k_N$2. A diagonal change of variables $k_N$3 converts the weight constraint into an $k_N$4 constraint at a determinant cost of $k_N$5.

The upper bound requires controlling the "tail" primes $k_N$6 via an iterated mixed-volume inequality (Lemma on $k_N$7 factors, applied through its corollary): each block of tail primes contributes a binomial factor whose logarithm sums to $k_N$8. Consequently both $k_N$9 and $M_k(N)=\mathcal P_N(k)$0 approximate $M_k(N)=\mathcal P_N(k)$1 with error vanishing after normalization by $M_k(N)=\mathcal P_N(k)$2.

Asymptotic evaluation of the model volumes

Each model body collapses, under projection onto block-sum coordinates, to a polytope $M_k(N)=\mathcal P_N(k)$3 with explicit fiber volumes (a Laplace-type integral with Stirling-controlled prefactors). Lemma model-asymp shows $M_k(N)=\mathcal P_N(k)$4, a variational functional combining entropy terms with a constrained supremum over $M_k(N)=\mathcal P_N(k)$5. Lagrange duality reduces the latter to a one-dimensional infimum over ordered parameters $M_k(N)=\mathcal P_N(k)$6; when the ratios $M_k(N)=\mathcal P_N(k)$7 are increasing and $M_k(N)=\mathcal P_N(k)$8, the unconstrained critical point $M_k(N)=\mathcal P_N(k)$9 is feasible, yielding the closed form involving $k\ge k_N$0.

Applying this to the prime-counting data (via the prime number theorem, $k\ge k_N$1), the lower and upper sums become Riemann sums for $k\ge k_N$2, which evaluates by integration by parts to exactly $k\ge k_N$3:

$k\ge k_N$4

A perturbation lemma handles the boundary block ($k\ge k_N$5), removing it at cost $k\ge k_N$6, negligible after taking $k\ge k_N$7 and $k\ge k_N$8.

Limitations and open questions

The main theorem is an asymptotic for the exponential scale of $k\ge k_N$9; no second-order term or polynomial-rate estimate for $\mathcal P_N$0 is obtained, and the effective threshold $\mathcal P_N$1 before which Khovanskii's polynomial behavior begins is bounded only crudely, via Granville–Smith–Walker's general bound $\mathcal P_N$2. The uniform-in-$\mathcal P_N$3 smooth asymptotic is established only for $\mathcal P_N$4, well below Mehdizadeh's $\mathcal P_N$5 range, and extending it is left open. The Riemann-sum argument fixes $\mathcal P_N$6 and $\mathcal P_N$7 before taking limits, so the method yields the leading constant but not finer asymptotics.

Conclusion

The paper resolves the large-dimension multiplication table problem at the level of the exponential scale: the leading Khovanskii coefficient satisfies $\mathcal P_N$8, with the constant emerging from an explicit integral tied to the distribution of primes above $\mathcal P_N$9. Combined with the smooth variant result $\pi(N)$0 uniformly in $\pi(N)$1 for $\pi(N)$2, the work connects Erdős's classical problem to Ehrhart-type theory of lattice polytopes and gives the first precise asymptotics in the varying-dimension regime.