Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$
For the analogous problem with Gaussian primes instead, click here.
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7$\begingroup$ The percolation threshold of the square lattice is roughly smaller than 0.6, while the density of squarefree Gaussian integers is approximately 0.66. So, if the squarefree Gaussian integers were randomly distributed, there would be an infinite component with probability 1. $\endgroup$– LeechLatticeDec 21, 2018 at 3:39
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