On the local discrepancy of (nα)-sequences

Let ω:=(xn)n⩾1 be any sequence of real numbers in the interval [0,1), N a positive integer and x∈[0,1). Then DN(ω,x):=∑n=1Nc[0,x)(xn)−Nx is called the local discrepancy of the sequence x1,…,xNat x. Here cM denotes the characteristic function of the set M. In this paper we investigate the case xn={nα}, when α denotes an irrational real number and {y} is the fractional part of the real number y. In the last few years the second author has sometimes been asked about the order of magnitude of DN(α,x) for x fixed and N large. It has been proved in Hecke (1922) [5] and in Kesten (1966) [6] that this quantity is bounded if and only if x is of the form {kα} for some integer k. Surprisingly, the general question can be reduced to the order of magnitude of sup0⩽x⩽1DN(α,x) and to inf0⩽x⩽1DN(α,x) for large N, a question which has been solved in Schoißengeier (1986) [9]. Finally, we apply our result to the case x=1/2.