If \(m, n\) are integers, \(1\leqslant m\leqslant n-1\) and \(s\in[m,n]\) (not necessarily an integer), then the \((m/s)\)-Hölder cube is the image of a Hölder continuous map \(f:[0,1]^m \to \mathbb{R}^n\) with exponent \((m/s)\). If \(m=1\), then the \((1/s)\)-Hölder cube is called a \((1/s)\)-Hölder curve. The symbol \(\mu \ \llcorner\ E\) means the restriction of the measure \(\mu\) to the set \(E \subset \mathbb{R}^n\), which means that \(\mu \ \llcorner\ E (F) = \mu (E \cap F)\) for all \(F \subset \mathbb{R}^n\). The main results of the paper are the following theorems:
Theorem A (Behavior at extreme lower densities). Let \(\mu\) be a Radon measure on \(\mathbb{R}^n\) and let \(s \in [1,n)\). Then the measure \[\underline{\mu}^s_0 :=\mu \ \llcorner\ \Bigg\{ x \in \mathbb{R}^n : \liminf_{r \downarrow 0} \frac{\mu(B(x,r))}{r^2} = 0\Bigg\}\] is singular to \((1/s)\)-Hölder curves. At the other extreme, the measure \[ \underline{\mu}^s_{\infty} := \mu\ \llcorner\ \Bigg\{ x \in \mathbb{R}^n : \int^1_0 \frac{r^s}{\mu(B(x,r))} \frac{dr}{r} < \infty \text{ and } \limsup_{r \downarrow 0} \frac{\mu(B(x, 2r))}{\mu(B(x,r))} < \infty \Bigg\} \] is carried by \((1/s)\)-Hölder curves, where “\(\underline{\mu}^s_{\infty}\) is carried by \((1/s)\)-Hölder curves” means that there exist countably many \((1/s)\)-Hölder curves \(\Gamma_i \subset \mathbb{R}^n\) such that \(\underline{\mu}^s_{\infty} (\mathbb{R}^n \setminus \bigcup_i \Gamma_i) = 0\) and “\(\underline{\mu}^s_0\) is singular to \((1/s)\)-Hölder curves” means that \(\mu (\Gamma) = 0\) for every \((1/s)\)-Hölder curve \(\Gamma \subset \mathbb{R}^n\).
Theorem B (Improvement to bi-Lipschitz curves). Let \(\mu\) be a Radon measure on \(\mathbb{R}^n\) and let \(t \in [0,1)\). Then the measure \(\mu^t_+\) is carried by bi-Lipschitz curves.
Theorem C (Improvement to \((m/s)\)-Hölder \(m\)-cubes). Let \(\mu\) be a Radon measure on \(\mathbb{R}^n\), let \(1 \leqslant m \leqslant n-1\) be an integer, let \(s \in [m,n)\), and let \(t \in [0,s)\). Then the measure \(\mu^t_+\) is carried by \((m/s)\)-Hölder \(m\)-cubes.
Here \[\mu^t_+ := \mu \ \llcorner\ \Bigg\{ x \in \mathbb{R}^n : 0 < \liminf_{r \downarrow 0} \frac{\mu (B(x,r))}{r^t} \leqslant \limsup_{r \downarrow 0} \frac{\mu (B(x,r))}{r^t} < \infty \Bigg\}.\]
To prove these theorems the authors consider (in Part I of the paper) several parametrization theorems concerning Hölder or bi-Lipschitz parametrizations of sets witch small Assouad dimension (Theorems 3.2 and 3.4). The definition and survey of the properties of this kind of dimension one can find in [P. Assouad, Bull. Soc. Math. Fr. 111, 429–448 (1983; Zbl 0597.54015)] or [J. Luukkainen, J. Korean Math. Soc. 35, No. 1, 23–76 (1998; Zbl 0893.54029)].
In Part II there are proofs of Theorems A, B and C. The proof of the first assertion of Theorem A uses the relationship between lower Hausdorff densities and packing measures. Theorems B and C follow from the connection between Hausdorff densities and Assouad dimension.
Similar considerations were carried out by M. A. Martín and P. Mattila from 1988 to 2000 for Hausdorff measures \(\mathcal{H}^s\) with \(0 < s < n\) not necessarily an integer [Trans. Am. Math. Soc. 305, No. 1, 293–315 (1988; Zbl 0643.28009); Math. Proc. Camb. Philos. Soc. 114, No. 1, 37–42 (1993; Zbl 0783.28005); Proc. Am. Math. Soc. 128, No. 9, 2641–2648 (2000; Zbl 0951.28005)]. The investigation of the measure – theoretic geometry of Euclidean sets of integral dimension goes back to A. S. Besicovitch in the 1920s and 1930s [Math. Ann. 98, 422–464 (1927; JFM 53.0175.04); Math. Ann. 115, 296–329 (1938; Zbl 0018.11302)].