Erdős--Turán Theorem and Eulerian Integers
Erik Füredi, Katalin Gyarmati
TL;DR
This work extends the Erdős–Turán framework from ordinary integers to the ring of Eulerian integers $E$, defining $ω_E$ and establishing a lower bound of order $\log|A|$ for $ω_E\big(\prod_{a,b\in A, a\neq b}(a+ρb)\big)$ when $A\subset E$ is finite and $ρ\in E$. Through the norm $N(a+bω)=a^2-ab+b^2$, these results yield corresponding bounds for $ω_{\mathbb{N}}$ on products like $\prod(a^2-ab+b^2)$ and $\prod(a^2+ab+b^2)$ with integer inputs, and motivate a general conjecture about two-variable polynomials. The authors prove a special case for polynomials of a structured form using Vandermonde determinants and a theorem of Győry–Sárközy–Stewart, and they provide computational evidence and a plan to extend to general homogeneous polynomials. The paper thus connects algebraic properties of Eulerian integers with classical prime-divisor phenomena, offering a robust Erdős–Turán-type toolkit in a nontrivial ring and laying groundwork for further generalizations and applications in Diophantine contexts.
Abstract
Our work is motivated by the fact that the norms of the Eulerian integers are related to the sums of form $a^2-ab+b^2$, providing a natural generalization for problems concerning products over sums or differences of integers. Let $E$ be the set of Eulerian integers. We define $ω_{\mathbb N}(x)$ as the number of distinct prime divisors of $x\in\mathbb N$, and $ω_E(x)$ as the number of distinct Euler prime divisors of $x\in E$. By the Erdős--Turán theorem, if $\mc A\subset\mathbb Z^{+}$ and $|\mathcal{A}|=3\cdot{2^{k-1}}$ ($k\in\mathbb{Z}^+$), then $ω_\mathbb{N}(\prod_{a,b\in\mathcal{A},a\neq{b}}(a+b))\geq{k+1}$. We prove that if $\mathcal{A} \subset E$ is a finite set and $ρ\in E$, then the value of $ω_E(\prod_{a,b \in \mathcal{A}, a \neq b}(a+ρb))$ has a lower bound of order $\log|\mathcal{A}|$. Consequently, we provide lower bounds for $\mathcal{A} \subset \mathbb{N}$ for both $ω_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2+ab+b^2))$ and $ω_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2-ab+b^2))$. We also give an upper bound for the minimum of $ω_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2+ab+b^2))$ with a computer program, if $|\mathcal{A}|\le 8$ and sets whose largest element is relatively small. Furthermore, using a Diophantine number theoretical lemma of Győry, Sárközy, and Stewart, we give a lower bound of order $\log|\mathcal{A}|$ for $ω_{\mathbb{N}}(\prod_{a \in \mathcal{A}, b \in \mathcal{B}}(f(a,b)))$ for a specific class of polynomials $f \in \mathbb{Z}[x,y]$ and finite sets $\mathcal{A}, \mathcal{B} \subset \mathbb{Z}$.