The Denjoy alternative for computable functions

The Denjoy-Young-Saks Theorem from classical analysis states that for an arbitrary function f : R → R, the Denjoy alternative holds outside a null set. This means that for almost every real x, either the derivative of f exists at x, or the derivative fails to exist in the worst possible way: the limit superior of the slopes around x equals +∞, and the limit inferior −∞. Algorithmic randomness allows us to define randomness notions giving rise to different concepts of almost everywhere. It is then natural to wonder which of these concepts corresponds to the almost everywhere notion appearing in the DenjoyYoung-Saks theorem. To answer this question Demuth investigated effective versions of the theorem. In his first variation, the function f is stipulated to be computable, and in the second one the function f is only Markov computable. For this second version, Demuth introduced a strong notion of randomness (stronger for example than Martin-Löf randomness) now known as Demuth randomness, which he proved to be sufficient to satisfy the Denjoy alternative for all Markov computable functions. In this paper, we in turn investigate these two effective theorems. We first show that the set of points that fulfill the Denjoy alternative for computable functions coincides with the set of computably random reals. We then show that the set of points that fulfill the Denjoy alternative for Markov computable functions is strictly bigger than the set of Demuth random reals — showing that Demuth’s sufficient condition was too strong — and moreover is incomparable with Martin-Löf randomness (meaning in particular that it does not correspond to any known set of random reals). To prove these two theorems, we study density-type theorems, such as the Lebesgue density theorem and obtain results of independent interest. We show for example that the classical notion of Lebesgue density can be characterized in an interesting way by the only very recently defined notion of difference randomness: x being difference random is equivalent to it being Martin-Löf random and having positive density in every effectively closed class in which x is contained. This is to our knowledge the first analytical characterization of difference randomness. We also consider the concept of porous points, a special type of Lebesgue non-density points that are well-behaved in the sense that the “density holes” around the point are continuous intervals whose ∗The second author is supported by a Feodor Lynen postdoctoral research fellowship by the Alexander von Humboldt Foundation. 1 ha l-0 06 26 27 5, v er si on 3 25 O ct 2 01 1 length follows a certain systematic rule. An essential part of our proof will be to argue that porous points of effectively closed classes can never be difference random.