We give a very short survey of the results on placing of points into the unit n-dimensional cube with mutual distances at least one. The main result is that into the 5-dimensional unit cube there can be
placed no more than 40 points.
We present a very short survey of known results and many new estimates and results on the maximum number of points that can
be chosen in the n-dimensional unit cube so that every distance between them is at least 1.
This paper generalizes earlier results on the behaviour of uniformly distributed sequences in the unit interval [0,1] to more general domains. We devote special attention to the most interesting special case [0,1]d. This will naturally lead to a problem in geometric probability theory, where we generalize results by Anderssen, Brent, Daley and Moran about random chord lengths in high-dimensional unit cubes, thereby answering a question by Bailey, Borwein and Crandall.
In dimension d = 2, the ‘log L’ term has power zero, which corresponds to a Theorem due to . The power on log L in dimension d ≥ 3 appears to be new, and supports a well-known conjecture on the L1 norm of DN. Comments on the discrepancy function in Hardy space also support the conjecture.