Let $A,B\subseteq \mathbb{N}$ such that for all large $N$$$\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}$$and$$\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.$$ Is it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\neq 0$ with $a_1,a_2\in A$ and $b_1,b_2\in B$?

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