The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

Based on the work of Mauldin and Williams [‘On the Hausdorff dimension of some graphs’, Trans. Amer. Math. Soc. 298(2) (1986), 793–803] on convex Lipschitz functions, we prove that fractal interpolation functions belong to the space of convex Lipschitz functions under certain conditions. Using this, we obtain some dimension results for fractal functions. We also give some bounds on the fractal dimension of fractal functions with the help of oscillation spaces.

The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx. 5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$. We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

Bilinear fractal interpolation surfaces were introduced by Ruan and Xu in 2015. In this paper, we present the formula for their box dimension under certain constraint conditions.

A borderline case function $f$ for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta \,>\,0$. On its support ${{I}_{0}}\,=\,{{\left[ 0,\,1 \right]}^{n}},\,f\left( x \right)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f\,=\,{{f}_{\beta }}\,\in \,{{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ if and only if $\alpha \,<\,\beta \,<\frac{n}{2}$ , while for $\beta \,=\,\alpha $, it was shown by Yue and Dafni that $f$ satisfies a John–Nirenberg inequality for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$. When $\beta \,\ne \,1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside ${{I}_{0}}$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta \,\in \,\left( 0,\,1 \right)$, $f$ is onto from ${{I}_{0}}$ to $\left[ -\frac{1}{1-{{2}^{-\beta }}},\,\frac{1}{1-{{2}^{-\beta }}} \right]$ , and the graph of $f$ has a non-integer fractal dimension $n\,+\,1\,-\beta$.