Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to...

Alfred Renyi, in a paper of 1962, A new approach to the theory of Engel's series, proposed a problem related to the growth of the elements of an Engel's series. In this paper, we reformulate and solve Renyi's problem for both, Engel's series and Pierce expansions

The approximants to regular continued fractions constitute "best approximations" to the numbers they converge to in two ways known as of the first and the second kind. This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a...