RESTRICTIONS OF HOLDER CONTINUOUS FUNCTIONS

For 0 < alpha < 1 let V (alpha) denote the supremum of the numbers v such that every alpha-Holder continuous function is of bounded variation on a set of Hausdorff dimension v. Kahane and Katznelson (2009) proved the estimate 1/2 <= V (alpha) <= 1/(2-alpha) and asked whether the upper bound is sharp. We show that in fact V (alpha) = max{1/2, alpha}. Let dim(H) and (dim) over bar (M) denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on V (a) is a consequence of the following theorem. Let {B(t) : t is an element of [0, 1]} be a fractional Brownian motion of Hurst index a. Then, almost surely, there exists no set A. [0, 1] such that (dim) over bar (M) A > max{1 - alpha, alpha} and B: A -> R is of bounded variation. Furthermore, almost surely, there exists no set A. [0, 1] such that (dim) over bar (M) A > 1 - alpha and B: A -> R is beta-Holder continuous for some beta > alpha. The zero set and the set of record times of B witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic a-Holder continuous functions. Finally, let {B(t) : t is an element of [0, 1]} be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set D subset of [0, 1] such that dim(H) D >= 1/3 and B: D -> R-2 is non-decreasing in each coordinate. It remains open whether 1/3 is best possible.