Hunt and Kaloshin (1999 Nonlinearity 12 1263–75) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d < k into for any N > 2k + 1, using a linear map L whose inverse is Hölder continuous with exponent α < (N − 2d)/N(1 + τ(X)/2), where τ(X) is the ‘thickness exponent’ of X. More recently, Ott et al (2006 Ergod. Theory Dyn. Syst. 26 869–91) studied the effect of such embeddings on the Hausdorff dimension of X, and showed that for ‘most’ linear maps , dH(L(X)) ⩾ min(N, dH(X)/(1 + τ(X)/2)). They also conjectured that ‘many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero’. In this paper we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X): we show that in both of the above results this can be used in place of the thickness exponent, and—appealing to results from the theory of approximate inertial manifolds—we prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt and Kaloshin. In particular, dev(X) = 0 for the attractor of the 2D Navier–Stokes equations with forcing f ∊ L2, while current results only guarantee that τ(X) = 0, when f ∊ C∞.