The Graham conjecture implies the Erdos-Turan conjecture

Abstract

Erdös and Turán once conjectured that any set $A\subset \mathbb{N}$ with ${\sum }_{a\in A}1/a=\infty$ should contain infinitely many progressions of arbitrary length $k\ge 3$. For the two-dimensional case Graham conjectured that if $B\subset \mathbb{N}\times \mathbb{N}$ satisfies $$\sum _{(x,y)\in B}\frac{1}{x^2+y^2}=\infty ,$$ then for any $s\ge 2$, $B$ contains an $s\times s$ axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some $s\ge 2$, then the Erdös-Turán conjecture is true for $k=2s-1$.

3 pages