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

TL;DR

This paper explores the relationships between Eulerian integers, prime divisors, and product sums, establishing lower bounds on the number of prime divisors for certain products and extending Erdős–Turán type results to Eulerian integers.

Contribution

It generalizes Erdős–Turán theorem bounds to Eulerian integers and related polynomial products, providing new lower bounds and computational results.

Findings

Lower bounds of order log|A| for prime divisors in products over Eulerian integers.

Extension of Erdős–Turán theorem to Eulerian integers and polynomial products.

Computational bounds for small sets and specific polynomial forms.

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 $\mathcal{A}, \mathcal{B} \subset \mathbb{Z}$.