On sums of fractional parts \(\{na+y\}\)
The authors obtain an explicit formula for the sum \[ C_m(\alpha,\gamma)= \sum_{1\leq k\leq m} (\{k\alpha+ \gamma\}-\textstyle{{1\over 2}}), \] where \(\alpha\) is an irrational. The formula is based on the continued fraction expansion of \(\alpha\), and the method is very related to \textit{J. Schoißengeier} [Acta Arith. 44, 241-279 (1984; Zbl 0544.10035); see also Monatsh. Math. 102, 59-77 (1986; Zbl 0613.10011); and Math. Ann. 296, 529-545 (1993; Zbl 0786.11043)] and \textit{C. Baxa} and \textit{J. Schoißengeier} [Acta Arith. 68, 281-290 (1994; Zbl 0828.11038)]. In the present paper, tight bounds for \(C_m(\alpha,\gamma)\) are established in terms of the generalized Zechendorf-expansion (= Ostrowski-expansion) related to \(\alpha\). Furthermore, an explicit formula is stated which immediately yields bounds for the discrepancy of the \((\{n\alpha\})^\infty_{n= 1}\) sequence.
scientific article; zbMATH DE number 1037291
scientific article