SYMBOLIC REPRESENTATION OF PIECEWISE-LINEAR FUNCTIONS ON THE UNIT INTERVAL AND APPLICATION TO DISCREPANCY

Let D(N)(U) be the star-discrepancy of a sequence U in the unit inter val. ''Self-similar sequences'' U are known for having a good discrepa ncy, i.e. L(U):= lim sup ND(N)(U)/ln N is finite, under some assumpti ons. We show in this paper that well-known techniques concerning subst itutions on finite alphabets and automata lead to some majorations of L(U), in a special case of self-similar sequences. The minimal obtaine d numerical value of L(U) is less than L(V), where V is the classical van der Corput sequence, but does not improve the lowest already known value.