Generic \(\frac12\)-discrepancy of \(\{n \theta+x\}\) (Q2439676)

The author studies the probabilistic behavior of discrepancy sums \[ S_{i}(x)=\sum^{i-1}_{j=0}\left(\chi_{[0,1/2)}- \chi_{[1/2,1)}\right)(x+j\chi), \] where \(\chi_{I}\) denotes the characteristic function of the interval I. Similar to the classical result of \textit{A. Khintchine} [Compos. Math. 1, 361--382 (1935; Zbl 0010.34101)], almost sure upper and lower bounds for the rate of divergence are given. The proofs essentially depend on mixing properties of the Gauss map and earlier work of the author [Acta Arith. 154, No. 1, 1--28 (2012; Zbl 1272.11092)].