Description and Exact Maximum and Minimum Values of the Remainder in the Problem of the Distribution of Fractional Parts

We obtain a description of the maximum and minimum values of the remainder in the problem of the distribution of fractional parts. The exact maximum and minimum values of the remainder are calculated; they are independent of the length of the bounded remainder interval. It is shown that these values can be computed in $O(m)$ operations. The remainder is described as a function of the argument $\alpha$.