It's somewhat a coincidence. In the paper, the dimension has to be larger than this constant K which is shown to be no larger than 1728. This is a fairly crude estimate that comes from multiplying a bunch of small numbers together, so the fact that it ends up being 12³ is not unimaginable. Then taking the dimension to be one larger than this estimate on K, we get 1729 = 12³ + 1³ (= 9³ + 10³).
However, 1728 isn't the minimum possible value for K. With more precise estimates, sketched in the paper, we can bring it down to 8 + ε. So assuming that this finer estimate works, using a 9 dimensional Fourier transform would also make the algorithm be O(n log(n)).
However, 1728 isn't the minimum possible value for K. With more precise estimates, sketched in the paper, we can bring it down to 8 + ε. So assuming that this finer estimate works, using a 9 dimensional Fourier transform would also make the algorithm be O(n log(n)).