A Metric Discrepancy Estimate for A Real Sequence

A general metrical result of discrepancy estimate related to uniform distribution is proved in this paper. It has been proven by J.W.S Cassel and P.Erdos & Koksma in [2] under a general hypothesis of (gn(x))￥n=1 that for every ε > 0, D(N, x) = O(N- (log N)5/2+ε) for almost all x with respect to Lebesgue measure. This discrepancy estimate was improved by R.C. Baker [5] who showed that the exponent 5/2+ε can be reduced to 3/2+ε in a special case where gn(x) = anx for a sequence of integers (an)￥n=1. This paper extends this result to the case where the sequence (an)￥n=1 can be assumed to be real. The lighter version of this theorem is also shown in this paper.