Erdős--Turán 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 $\ddot{I}{‰}_{\mathbb{N}}(x)$ as the number of distinct prime divisors of $x\in \mathbb{N}$, and $\ddot{I}{‰}_E(x)$ 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 $\ddot{I}{‰}_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\ne b}(a+b))\ge k+1$. We prove that if $\mathcal{A}\subset E$ is a finite set and $\ddot{I}\in E$, then the value of $\ddot{I}{‰}_E({\prod }_{a,b\in \mathcal{A},a\ne b}(a+\ddot{I}b))$ has a lower bound of order $\log |\mathcal{A}|$. Consequently, we provide lower bounds for $\mathcal{A}\subset \mathbb{N}$ for both $\ddot{I}{‰}_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\ne b}(a^2+ab+b^2))$ and $\ddot{I}{‰}_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\ne b}(a^2-ab+b^2))$. We also give an upper bound for the minimum of $\ddot{I}{‰}_{\mathbb{N}}({\prod }_{a,b\in \mathcal{A},a\ne b}(a^2+ab+b^2))$ 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 $\ddot{I}{‰}_{\mathbb{N}}({\prod }_{a\in \mathcal{A},b\in \mathcal{B}}(f(a,b)))$ for a specific class of polynomials $f\in \mathbb{Z}[x,y]$ and finite sets $\mathcal{A},\mathcal{B}\subset \mathbb{Z}$.

Submitted to Acta Arithmetica