Dynamically defined sequences with small discrepancy

We study the problem of constructing sequences (x on [0, 1] in such a way that D*N = suo 0 <= x <= 1 vertical bar#{1 <= i <= N : Xi <= x}/N-x broken vertical bar is small. A result of Schmidt shows that for all sequences sequences (x,,),T on [0, 1] we have D'i'v > (log N)N-1 for infinitely many N, several classical constructions attain this growth. We describe a type of uniformly distributed sequence that seems to be completely novel: given (xi,. " ", xN d, we construct xN in a greedy manner xN = arg min Sigma k=1N-1 1 - log (s sin (pi vertical bar x - xk vertical bar)). We prove that D*N < (log N)N-1/2 and conjecture that 1)k < (log N)N-1. Numerical examples illustrate this conjecture in a very impressive manner. We also establish a discrepancy bound (log N)a. N -1I2 for an analogous construction in higher dimensions and conjecture it to be D < (log N)d N-1.