Kloostermanʹs uniformly distributed sequence

Applying the theory of uniform distribution, especially the Erdös–Turán–Koksma inequality and the Koksma–Hlawka inequality, to the two-dimensional Kloosterman sequence (aj/n,aj*/n), j=1,2,…, (n) (where ajaj*≡1 (mod n), aj,aj* [1,n] and (n) is the Euler function) we find an estimation for the discrepancy D (n)* of this sequence and an error term for the Kth moment, K=1,2,…, of the sequence of distances aj/n−aj*/n as where the Hardy–Krause variation V(x−yK)=4 and the discrepancy (d(n) is the divisor function). From known estimates of Kloostermanʹs sum immediately follows uniform distribution of the sequence (aj/n,aj*/n), j=1,2,…, (n), as n→∞ which directly implies that the related sequence of distances aj/n−aj*/n has the asymptotic distribution function g(x)=2x−x2. For a general sequence of points (xj,yj), j=1,2,…,N, in the unit square [0,1)2 we find an approximation of the discrepancy of xj−yj by the discrepancy of (xj,yj) which gives These results improve and unify some of Zhangʹs results published in Zhang (J. Number Theory 52 (1995) 1–6; J. Number Theory 61 (1996) 301–310, Acta Math. Hungar. 76 (1997) 17–30) from the point of view of uniform distribution theory.