An example of a function which is Denjoy integrable but not khinchin summable

The following result is proven: if ξ is an irrational number “anomalously badly“ approximable by rationals, then there are functions which are not Khinchin ξ-summable but which are Denjoy integrable. Let I be the interval 0 ≤ x ≤ 1, and let ξ be an irrational, 0 < ξ< 1. Let T _ξ denote the transformation of I into itself defined as follows: $$T_\xi x = \left\{ {\begin{array}{*{20}c} {x + \xi ,ifx + \xi \in I;} \\ {x + \xi - 1} \\ \end{array} } \right.$$ otherwise.