The Main Conjectures
Conjecture 1. There is a constant cs depending only on the dimension s, such that
for any finite point set {x1,...,xN}.
Conjecture 2. There is a constant c's depending only on s, such that
for at least one N for any infinite sequence x1,x2,x3,....
These conjectures are equivalent. They have been proved for s ≤ 2 by W. M. Schmidt. In higher dimensions, the corresponding problem is still open. The best-known lower bounds are due to K. F. Roth.
Read more about this topic: Low-discrepancy Sequence
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