Erdős--Turán Theorem and Eulerian Integers

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 ${\omega }_{\mathbb{N}}(x)$ as the number of distinct prime divisors of $x\in \mathbb{N}$, and ${\omega }_E(x)$ as the number of distinct Euler prime divisors of $x\in E$. By the Erdős–Turán theorem, if $\text{mc}A\subset {\mathbb{Z}}^{+}$ and $\mathrm{\mid }\mathcal{A}\mathrm{\mid }=3\cdot 2^{k-1}$ ($k\in {\mathbb{Z}}^{+}$), then ${\omega }_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\mathrm{\ne }b}(a+b))\ge k+1$. We prove that if $\mathcal{A}\subset E$ is a finite set and $\rho \in E$, then the value of ${\omega }_E({\prod }_{a,b\in \mathcal{A},a\mathrm{\ne }b}(a+\rho b))$ has a lower bound of order $\log \mathrm{\mid }\mathcal{A}\mathrm{\mid }$. Consequently, we provide lower bounds for $\mathcal{A}\subset \mathbb{N}$ for both ${\omega }_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\mathrm{\ne }b}(a^2+ab+b^2))$ and ${\omega }_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\mathrm{\ne }b}(a^2-ab+b^2))$. We also give an upper bound for the minimum of ${\omega }_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\mathrm{\ne }b}(a^2+ab+b^2))$ with a computer program, if $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\le 8$ 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 $\log \mathrm{\mid }\mathcal{A}\mathrm{\mid }$ for ${\omega }_{\mathbb{N}}({\prod }_{a\in \mathcal{A},b\in \mathcal{B}}(f(a,b)))$ for a specific class of polynomials $f\in \mathbb{Z}[x,y]$ and finite sets $\mathcal{A},\mathcal{B}\subset \mathbb{Z}$.