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 $(g_n (x))_{n = 1}^\infty$ that for every $\varepsilon>0$, $$D(N,x) = O(N^{\frac{{ - 1}}{2}} (\log N)^{\frac{5}{2} + \varepsilon } )$$ for almost all $x$ with respect to Lebesgue measure. This discrepancy estimate was improved by R.C. Baker [5] who showed that the exponent $\frac{5}{2} + \varepsilon$ can be reduced to $\frac{3}{2} + \varepsilon$ in a special case where $g_n (x) = a_n x$ for a sequence of integers $(a_n )_{n = 1}^\infty$. This paper extends this result to the case where the sequence $(a_n )_{n = 1}^\infty$ 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