A note on cube-free problems
Yuchen Meng
TL;DR
The paper studies $d$-cube-free subsets of the cyclic group $\mathbb{Z}_N$ with $d\mid N$, proving the conjectured density bound $|A|\leq \frac{d-1}{d}N$ in several important cases and establishing tightness via the residue class example $A=\{a: a\equiv 1,\dots, d-1\pmod{d}\}$. It introduces a constructive matrix- or layer-based framework and leverages the Cauchy–Davenport theorem, along with double counting, to bound the size of cube-free sets. The work further explores related problems including $(x,dx)$-free and $(x,2x,\ldots, dx)$-free sets, deriving upper bounds in $[N]$ and $\mathbb{Z}_N$, and demonstrates the conjecture in notable cases: $d=3$, when $d$ is the smallest prime factor of $N$, and when $N$ is a prime power. These results advance understanding of cube-free phenomena in additive combinatorics and suggest avenues for extending the bounds to broader families of $N$ and $d$.
Abstract
Eberhard and Pohoata conjectured that every $3$-cube-free subset of $[N]$ has size less than $2N/3+o(N)$. In this paper we show that if we replace $[N]$ with $\mathbb{Z}_N$ the upper bound of $2N/3$ holds, and the bound is tight when $N$ is divisible by $3$ since we have $A=\{a\in \mathbb{Z}_N:a\equiv 1,2\pmod{3}\}.$ Inspired by this observation we conjecture that every $d$-cube-free subset of $\mathbb{Z}_N$ has size less than $(d-1)N/d$ where $N$ is divisible by $d$, and we show the tightness of this bound by providing an example $B=\{b\in\mathbb{Z}_N:b\equiv 1,2,\ldots,d-1\pmod{d}\}$. We prove the conjecture for several interesting cases, including when $d$ is the smallest prime factor of $N$, or when $N$ is a prime power. We also discuss some related issues regarding $\{x,dx\}$-free sets and $\{x,2x,\ldots,dx\}$-free sets. A main ingredient we apply is to arrange all the integers into some square matrix, with $m=d^s\times l$ having the coordinate $(s+1,l-\lfloor l/d\rfloor)$. Here $d$ is a given integer and $l$ is not divisible by $d$.