Dichotomy results for the L1 norm of the discrepancy function

It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L2 norm. In dimension d=2 this fact has been established by Halász, 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 L1 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.