On the derivative of the Minkowski question mark function ?(x)

Abstract

Let x = [0; a1, a2, . . .] be the regular continued fraction expansion of an irrational number x ∈ [0, 1]. For the derivative of the Minkowski function ?(x) we prove that ? (x) = +∞, provided that lim sup t→∞ a 1 +···+a t t < κ1 = 2 log λ 1 log 2 = 1.388 + , and ? (x) = 0, provided that lim inf t→∞ a 1 +···+a t tConstants κ1, κ2 are the best possible. It is also shown that ? (x) = +∞ for all x with partial quotients bounded by 4.

The Minkowski question mark function: explicit series for the dyadic period function and moments

“…A by far not complete overview of the papers written about the Minkowski question mark function or closely related topics (Farey tree, enumeration of rationals, Stern's diatomic sequence, various 1-dimensional generalizations and generalizations to higher dimensions, statistics of denominators and Farey intervals, Hausdorff dimension and analytic properties) can be found in [1]. These works include [5], [6], [8], [9], [10], [12], [13] (this is the only paper where the moments of a certain singular distribution, a close relative of F (x), were considered), [11], [14], [16], [18], [20], [24], [25], [26], [27], [28], [29], [30], [31], [33]. The internet page [36] contains an up-to-date and exhaustive bibliographical list of papers related to the Minkowski question mark function.…”

Methods for estimating continuants

“…For K l,n , the paper [4] contains an inequality which is close to (38). Equality (38) for k l,n can be proved in a similar way.…”

“…The basis of induction applies by Remark 3. Lemma 4 is proved.Suppose, for a certain i ≥ 0 and for sequences {b j } and {c j } defined in(4), that…”

Existence theory for semilinear evolution inclusions involving measures

“…the Minkowski question mark function (singular, continuous and strictly increasing, but not absolutely continuous; cf. [25], for instance) we get strong counterexample for possibility of use of earlier results (note that there are several constructions for a measure that, when integrated, yields the question mark function). Singular continuous functions were not covered by earlier results.…”