Pseudo random sequences generated by piecewise linear maps from the view point of dynamical system (Q5938376)

The discrepancy \(D(M) \) of the sequence \( S= \{x_1,x_2,x_3, \dots\}\), \(x_i \in \langle 0,1\rangle\) is defined as \(\sup_J D_J (M)\) where NEWLINE\[NEWLINED_J(M)= \left|{|{J}|-\frac{ \#\{x_i \in J: i \leq M\}}{M}}\right|NEWLINE\]NEWLINE The supremum is taken over all subintervals \(J \subset \langle 0,1\rangle\) and \(|J|\) is the Lebesgue measure of \(J\). \(S\) is of low discrepancy if \(D(M) = O(\log M/M)\). The discrepancy of van der Corput sequences (CS) is studied. CS is often of low discrepancy. The paper presents a method of the construction of CS. Using results of the ergodic theory and picewise linear Markov transformations the author presents upper bounds for the discrepancies of some CS's, a condition for a CS not to be of low discrepancy and sufficient and necessary conditions under which a CS is of low dicrepancy.