Let {W(t):t [ges ] 0} denote a Wiener process, and set [Sscr ] for the unit ball of the reproducing kernel Hilbert space pertaining to the restriction of W on [0,1], with Hilbert norm [mid ] · [mid ]H. Gorn and Lifshits [8] have shown that, whenever f ∈ [Sscr ] fulfills [mid ] f [mid ]H = 1 and has Lebesgue derivative of bounded variation, the rate of clustering of (2h log(1/h))−½(W(t + h·) − W(t)) to f is of the order O((log(1/h))−2/3. In this paper, we show that the set of exceptional points in [0,1] where this rate is reached constitutes a random fractal whose Hausdorff–Besicovitch measure is evaluated.