The Erdős–Turán Conjecture in the positive rational numbers
Volume 152 / 2018
Abstract
The well known Erdős–Turán conjecture says that if $A$ is a subset of the natural numbers such that every sufficiently large integer can be represented as a sum of two integers of $A$, then the number of such representations cannot be bounded. We prove that this is false in the positive rational numbers $\mathbb {Q^+}$. For any $A\subseteq \mathbb {Q^+}$ and any $\alpha \in \mathbb {Q^+}$, let $R(A, +, \alpha )$, $R(A, -, \alpha )$, $R(A, \cdot , \alpha )$ and $R(A, \div , \alpha )$ denote the numbers of solutions of $\alpha =a+b $ $ (a\le b)$, $\alpha =a-b$, $\alpha =ab$ $ (a\le b)$ and $\alpha =a/b$, with $a, b\in A$, respectively. We prove that there exists a subset $A$ of $\mathbb {Q^+}$ such that, for any $\alpha \in \mathbb {Q^+}\setminus \{ 1\} $, we have $R(A, +, \alpha )=1$, $ R(A, -, \alpha )=1$, $R(A, \cdot , \alpha )=1$, $R(A, \div , \alpha ) =1$, as well as $R(A, +, 1 )=1$, $R(A, -, 1 )=1$ and $R(A, \cdot , 1)=1$. We also prove similar results in $\mathbb {Q^+}\cap (1, \infty )$ and $\mathbb {Q^+}\cap (0, 1)$.
