Erd\H{o}s--Tur\'an Theorem and Eulerian Integers

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 $\omega_{\mathbb N}(x)$ 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 Erd\H{o}s--Tur\'an theorem, if $\mc A\subset\mathbb Z^{+}$ and $|\mathcal{A}|=3\cdot{2^{k-1}}$ ($k\in\mathbb{Z}^+$), then $\omega_\mathbb{N}(\prod_{a,b\in\mathcal{A},a\neq{b}}(a+b))\geq{k+1}$. We prove that if $\mathcal{A} \subset E$ is a finite set and $\rho \in E$, then the value of $\omega_E(\prod_{a,b \in \mathcal{A}, a \neq b}(a+\rho b))$ has a lower bound of order $\log|\mathcal{A}|$. Consequently, we provide lower bounds for $\mathcal{A} \subset \mathbb{N}$ for both $\omega_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2+ab+b^2))$ and $\omega_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2-ab+b^2))$. We also give an upper bound for the minimum of $\omega_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2+ab+b^2))$ 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 Gy\H{o}ry, S\'ark\"ozy, and Stewart, we give a lower bound of order $\log|\mathcal{A}|$ 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 $\mathcal{A}, \mathcal{B} \subset \mathbb{Z}$.