On Irregularities of Distribution II

Let a=(a₁, a₂, a₃, ...) be an arbitrary infinite sequence inU=0, 1). Let Van der Corput 5] conjectured that d(a, n) (n=1, 2, ...) isunbounded, and this was proved in 1945 by van Aardenne-Ehrenfest1]. Later she refined this 2], obtaining for infinitely many n. Here and later c₁, c₂, ... denote positiveabsolute constants. In 1954, Roth 8] showed that the quantity is closely related to the discrepancy of a suitable point setin U².