A Metric Discrepancy Estimate for A Real Sequence

Abstract

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))1n=1 that for every " > 0, D(N, x) = O(N −1 2 (logN) 5 2+") 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 + " can be reduced to 3 2 + " in a special case where gn(x) = anx for a sequence of integers (an)1n=1. This paper extends this result to the case where the sequence (an)1n=1 can be assumed to be real. The lighter version of this theorem is also shown in this paper. Keywords Discrepancy, uniform distribution, Lebesgue measure, almost everywhere