Continued fractions with bounded even-order partial quotients

The paper is concerned with continued fractions having bounded even-order partial quotients. We demonstrate that every real number can be written as a sum of two continued fractions whose even-order partial quotients are equal to 1, and every positive number can be written as a product of two such continued fractions. Then we study the Hausdorff dimension of the set of continued fractions whose even-order partial quotients are all equal to a given positive integer c. Taking in particular c=1, we show that the set of continued fractions with even-order partial quotients equal to 1 has the Hausdorff dimension between 0.732 and 0.819.