Erd\"{ o}s and Tur\'{ a}n 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 $k\geq3$. For the two-dimensional case Graham conjectured that if $B\subset \mathbb{ N}\times\mathbb{ N}$ satisfies $$\sum\limits_{ (x,y)\in B}\frac{ 1}{ x^2+y^2}=\infty,$$ then for any $s\geq2$, $B$ contains an $s\times s$ axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some $s\geq2$, then the Erd\"{ o}s-Tur\'{ a}n conjecture is true for $k=2s-1$.
Categories: math.NT