Dichotomy results for the norm of the discrepancy function

It is a well-known conjecture in the theory of irregularities of distribution that the L 1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L 2 norm. In dimension d = 2 this fact has been established by Halلsz, while in higher dimensions the problem is wide open. In this note, we establish a series of dichotomy-type results which state that if the L 1 norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be large in some other function space.