Conformal Dimension of Measures and Quasisymmetric Dimension Reduction

Abstract: We prove that the conformal dimension of every locally finite Borel measure is either zero or infinite. The main ingredient is a quasisymmetric dimension-reduction theorem: every full-support probability measure of finite Hausdorff dimension on a separable metric space admits quasisymmetrically equivalent metrics in which its Hausdorff dimension is arbitrarily small. In particular, every locally finite Borel measure on a doubling metric space has conformal dimension zero. We also prove that for every $n\geq 1$ and $p&gt;0$ there are a metric space $X$, a quasisymmetric homeomorphism $f:[0,1]^n\to X$, and a Borel set $E\subset[0,1]<sup>n$ such that $\dim_H f(E)\leq p$ and $\dim_H([0,1]^n\setminus E)\leq n-1+p$. The second bound is sharp up to $p$: if $\dim_H f(E)&lt;1$, then $\dim_H([0,1]^n\setminus E)\geq n-1$.

• The paper proves that the conformal dimension of every locally finite Borel measure is either 0 or +∞, resolving a question of Bate and Orponen and showing that measures on doubling spaces always have conformal dimension zero.
• The paper develops a randomized weighted hyperbolic-filling construction that produces quasisymmetrically equivalent metrics with Hausdorff measure dimension below any prescribed p>0, including noncompact and noncomplete spaces.
• The paper quantitatively strengthens Romney’s singular-map theorem by constructing maps with H^d(f(E))≤p and dim_H([0,1]^n\E)≤n−1+p, while proving a topological obstruction that prevents reducing the complement below dimension n−1 when dim_H f(E)<1.

This paper by Hua Qiu and Qi Wang resolves a question of Bate and Orponen on the conformal dimension of measures, proving that this invariant is always either zero or infinite, and establishes a sharp quantitative refinement of Romney's theorem on singular quasisymmetric maps (2608.12873).

The dichotomy for conformal dimension of measures

The conformal dimension $\mathrm{Cdim}\,X$ of a metric space, introduced by Pansu, is the infimum of Hausdorff dimensions over quasisymmetric images of $X$. Bate and Orponen extended this to measures: for a locally finite Borel measure $\mu$ with support $S$, the conformal dimension $\mathrm{Cdim}\,\mu$ is the infimum of $\mathcal H^d(f_\#\mu)$ over quasisymmetric homeomorphisms $f:S\to Y$. They observed that the measure-theoretic analogue of the Bishop–Tyson minimality phenomenon can fail dramatically — for every $n\ge2$ and $0$\mathcal H^{1+s}$ to $X$0 has conformal dimension zero — and asked whether positive finite values occur.

The main theorem answers negatively: for every locally finite Borel measure $X$1 on any metric space, $X$2. The proof splits by separability of the support. If $X$3 is nonseparable, any full-measure Borel set under a quasisymmetric image would have finite Hausdorff dimension, hence be separable, forcing $X$4 separable — a contradiction; so the value is $X$5. If $X$6 is separable, local finiteness makes $X$7 $X$8-finite, and an equivalent probability measure $X$9 (with strictly positive density) shares all null sets with $\mu$0. Applying the dimension-reduction theorem to $\mu$1 then drives $\mu$2 below any prescribed $\mu$3.

An immediate corollary is that every locally finite Borel measure on a doubling metric space has conformal dimension zero, via Assouad's snowflake embedding into some $\mu$4. In particular, Lebesgue measure $\mu$5 has conformal dimension zero for all $\mu$6, and Question 2 of Bate–Orponen concerning $\mu$7-dimensional Hausdorff measure on $\mu$8 has a negative answer. The authors also give an independent proof of the doubling corollary combining Assouad's embedding with Romney's singular map construction and a Fubini/translation argument showing that a generic translate avoids the exceptional set.

Without doubling, both values occur: counting measure on an uncountable discrete space has infinite conformal dimension, while a product ultrametric space $\mu$9 with its natural product measure has conformal dimension zero, since the snowflakes $S$0 are quasisymmetrically equivalent and reduce the dimension below $S$1.

The quasisymmetric dimension-reduction theorem

The engine behind the main result is: if $S$2 is a full-support Borel probability measure of finite Hausdorff dimension on a separable metric space, then for every $S$3 there is a quasisymmetrically equivalent metric in which $S$4. The construction adapts the weighted hyperbolic-filling technique of Miller and Tian (Miller et al., 25 Mar 2026, Miller et al., 28 Apr 2026) with two extensions: the vertex sets at each scale may be countably infinite (since $S$5 need not be compact), and the metric is defined directly on $S$6 via chain costs rather than on a graph boundary.

The filling consists of nested maximal $S$7-separated sets, horizontal adjacency at distance $S$8, and vertical parent edges. Vertex weights $S$9 are built recursively from an admissible assignment $\mathrm{Cdim}\,\mu$0, chosen so that every horizontal chain crossing the annulus between radii $\mathrm{Cdim}\,\mu$1 and $\mathrm{Cdim}\,\mu$2 has total cost at least one. A key lemma shows every finite path from $\mathrm{Cdim}\,\mu$3 to $\mathrm{Cdim}\,\mu$4 costs at least $\mathrm{Cdim}\,\mu$5, where $\mathrm{Cdim}\,\mu$6 is the weight of the deepest common horizontal-neighbor ancestor; this prevents degeneration of the resulting chain metric $\mathrm{Cdim}\,\mu$7, which is shown to be quasisymmetrically equivalent to $\mathrm{Cdim}\,\mu$8, with ball diameters controlled by a product $\mathrm{Cdim}\,\mu$9 of per-scale factors $\mathcal H^d(f_\#\mu)$0 or $\mathcal H^d(f_\#\mu)$1.

The weights are randomized using independent random Borel partitions at each scale, constructed by Voronoi-type cells around i.i.d. samples from $\mathcal H^d(f_\#\mu)$2 with random radius. A partition lemma gives the bound

$\mathcal H^d(f_\#\mu)$3

so that, choosing $\mathcal H^d(f_\#\mu)$4, Kolmogorov's strong law plus Fubini yields a deterministic choice of partitions for which almost every point sees the favorable factor $\mathcal H^d(f_\#\mu)$5 at all but a fraction $\mathcal H^d(f_\#\mu)$6 of scales, where $\mathcal H^d(f_\#\mu)$7. Combining with the mass lower bound $\mathcal H^d(f_\#\mu)$8 along a subsequence forces the lower local dimension under $\mathcal H^d(f_\#\mu)$9 below $f:S\to Y$0, and $f:S\to Y$1 is then taken arbitrarily small. Unbounded spaces are handled by sphericalization (adding a point at infinity, with explicit bi-Lipschitz comparisons on bounded sets), incomplete spaces by completion.

Sharp singular quasisymmetric maps

The second main theorem refines Tukia's one-dimensional result and Romney's higher-dimensional construction: for every $f:S\to Y$2 and $f:S\to Y$3 there exist a metric space $f:S\to Y$4, a quasisymmetric homeomorphism $f:S\to Y$5, and a Borel set $f:S\to Y$6 with

$f:S\to Y$7

Romney's original theorem achieved only $f:S\to Y$8; the new argument controls the dimension of the complement explicitly. The proof uses Romney's conformal weights on the $f:S\to Y$9-adic grid, where cubes descending through the central region gain a factor $n\ge2$0. A cube is called $n\ge2$1-good if it visits the central region at least $n\ge2$2 times in $n\ge2$3 steps; an entropy estimate bounds the number of bad cubes, giving $n\ge2$4, while large $n\ge2$5 forces $n\ge2$6 via the diameter estimate $n\ge2$7.

The bound $n\ge2$8 is sharp up to the additive term: the authors prove that whenever $n\ge2$9 is a homeomorphism and $0

Limitations and open questions

The results are stated for the upper (full-measure) definition of the Hausdorff dimension of a measure. The authors note they do not know whether the dichotomy theorem persists when conformal dimension is defined via the lower Hausdorff dimension $0$\mathcal H^{1+s}$0 for each fixed $\mathcal H^{1+s}$1.

Conclusion

The paper establishes a complete dichotomy — conformal dimension of a locally finite Borel measure is either $\mathcal H^{1+s}$2 or $\mathcal H^{1+s}$3 — resolving both questions of Bate and Orponen, with the doubling case yielding conformal dimension zero universally. Technically, it extends weighted hyperbolic filling constructions to noncompact settings via randomized partitions adapted to the measure, and delivers a sharp quantitative version of singular quasisymmetric maps in all dimensions, with a matching topological obstruction showing the bound $\mathcal H^{1+s}$4 cannot be substantially improved.