Lower bounds for the discrepancy of triples of inversive congruential pseudorandom numbers with power of two modulus

This paper deals with the inversive congruential method with power of two modulusm for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers. It is shown that there exist parameters in the inversive congruential method such that the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude at leastm−1/3. The method of proof relies on a detailed analysis of certain rational exponential sums.