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 interval. ''Self-similar sequences'' U are known for having a good discrepancy, i.e. L(U):= lim sup ND(N)*(U)/ln N is finite, under some assumptions. We show in this paper that well-known techniques concerning substitutions on finite alphabets and automata lead to some majorations of L(U), in a special case of self-similar sequences. The minimal obtained 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.