# Why does $l_0$ appear in this statement of the Furstenberg–Katznelson–Weiss theorem?

In Terence Tao's paper Exploring the toolkit of Jean Bourgain is stated:

Theorem 3.1 (Furstenberg–Katznelson–Weiss theorem, qualitative version). Let $$A\subset\Bbb R^2$$ be a measurable set whose upper density $$δ∶=\limsup_{R→∞}\frac{|A∩\mathrm B(0,R)|}{|\mathrm B(0,R)|}$$is positive. Then there exists $$l_0$$ such that, for all $$l≥l_0$$, there exist $$x, y∈A$$ with $$|x−y|\geqslant l$$.

In this paper, in addition to the usual meaning, $$|\centerdot|$$ denotes the Lebesgue measure of a subset of $$\Bbb R^2$$, and I interpret $$\mathrm B(0,R)$$ as the (open) ball of radius $$R$$ and centered at the origin in $$\Bbb R^2$$.

A minor point is that the conclusion of the theorem is trivial unless $$l_0$$ is required to be positive. That granted, though, it seems that $$l_0$$ is entirely redundant. Thus, why does the last sentence of the theorem not stay simply “Then, for all $$l$$, there exist $$x, y∈A$$ with $$|x−y|\geqslant l$$.”? Moreover, I don't even see the need for $$\delta$$ to be positive. All that is necessary is for $$A$$ to be unbounded (e.g. $$A=\Bbb N\times\{0\}$$), in which case the theorem's conclusion is pretty well tautologous.

I guess that there is a crucial typo somewhere, whose correction would make the theorem nontrivial. But I can't see what that could be.

• I believe it's $|x-y| = l$ that might make it the correct statement. – asahay Nov 13 '20 at 11:55
• @asahay : Thanks, that would make sense. If you posted that as your answer, I could accept it. – John Bentin Nov 13 '20 at 12:16

I looked up the paper in which Furstenberg-Katznelson-Weiss proved their result. The correct statement has $$|x-y| = \ell$$ instead of $$|x-y| \geq \ell$$.