> The methods of creating fully deterministic low discrepancy quasirandom sequences in one dimension are extremely well studied, and generally solved.
Are you referring to something like quadratic residue [1]? It's basically that the sequence x^2 mod P where P is prime appears to be somewhat random, and it goes through the whole sequence before repeating. But it's obviously finite, and it's not perfect [2]. I'm mostly just curious how it relates.
My intent when I wrote that phrase was two-fold. Firstly to emphasise the large body of work that was done by Weyl, Kronecker, Hurwitz et al relating to the equidistribtuion theorem, badly approximately numbers and diophantine approximation. Armed with this knowledge it is easily proved that the recurrrence relation using the golden ratio mod 1 is optimal using almost any measure.
Although some other readers on HN may be able to see a connection between one dimensional quasirandom sequences and quadratic residue, unfortunately I can’t immediately see an explicit one. Sorry!
My second intent was to contrast this with how little is provably known for higher dimensions. To my knowledge, thanks to the phenomenal work by Halton, Sobol, Niederreiter et al, we know that many of the contemporary sequences are optimal in the limiting Big O sense, but there are no proofs even for d=2 for optimality for any of the sequences for general finite values of n.
Are you referring to something like quadratic residue [1]? It's basically that the sequence x^2 mod P where P is prime appears to be somewhat random, and it goes through the whole sequence before repeating. But it's obviously finite, and it's not perfect [2]. I'm mostly just curious how it relates.
[1]: https://en.wikipedia.org/wiki/Quadratic_residue [2]: https://arxiv.org/ftp/arxiv/papers/1612/1612.05852.pdf