Product representations of perfect powers

TL;DR

This work studies the extremal problem of selecting a largest subset of $[N]$ with no product of $k$ distinct elements equal to a perfect $d$-th power. It proves that for prime power $d$ and sufficiently large $N$, the extremal size is exactly $\rho_d(N)=\sum_{k=1}^{d-1}\pi\left(\frac{N}{k}\right)$, and in general gives a bound $\rho_d(N)=\sum_{k=1}^{d-1}\pi\left(\frac{N}{k}\right)+O_d\left(\pi(\sqrt{N})\right)$. The method weaves Davenport's constant for finite abelian groups into a structural decomposition of candidate sets, encoding prime-exponent vectors modulo $d$ as zero-sum-free sequences in $\mathbb{Z}_d^{m}$ to bound the size of the $d$-smooth component, and refining this with a $\sqrt{N}$-based partition to control the remaining part. A matching construction shows the exact value for prime powers when $N\ge 2^d p_d$, resolving a question of Verstraëte in this regime; the paper also discusses potential improvements to the error term and threshold for general $d$.

Abstract

Let $ρ_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right) +O_d(π(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right)$. This answers a question of Verstraëte.

Theorems & Definitions (10)

• Theorem 1.1: Verstraëte
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• proof : Proof of Theorem \ref{['upper1']}
• proof : Proof of Theorem \ref{['upper2']}
• proof : Proof of Theorem \ref{['thm-exact']}