A question of Alon and Erdős [AlEr85], who proved $\lvert A\rvert \geq N^{2/3-o(1)}$ is possible (via a random subset), and observed that \[\lvert A\rvert \ll \frac{N}{(\log N)^{1/4}},\] since (as shown by Landau) the density of the sums of two squares decays like $(\log N)^{-1/2}$. The lower bound was improved to \[\lvert A\rvert \gg N^{2/3}\] by Lefmann and Thiele [LeTh95].