Quantitative Distance Bounds

• Quantitative Distance Bounds are explicit, parameter- and dimension-dependent estimates that control geometric, analytic, or probabilistic distances based on intrinsic properties.
• They provide concrete comparison scales in metric geometry, optimal transport, and combinatorial settings by establishing near-isometries and sensitivity measures.
• These bounds underpin stability analyses and performance guarantees in high-dimensional geometry, coding theory, and quantum information applications.

Quantitative distance bounds are explicit, dimension- and parameter-dependent estimates that control various geometric, analytic, or probabilistic distances in terms of intrinsic properties of the spaces, objects, or measures involved. These bounds arise across metric geometry, group actions, functional and probabilistic inequalities, combinatorics, coding theory, and quantum information, and provide canonical means to compare or approximate distances and to measure stability or sensitivity.

1. Distances and Quantitative Bounds in Metric Geometry and Group Actions

A central class of results concerns orbits of abelian group actions in geodesic spaces. Given a cocompact, properly discontinuous free action by $\Gamma \cong \mathbb{Z}^n$ on a length space $(X, d)$, the stable norm $\|\cdot\|_{st}$ on $\Gamma \otimes \mathbb{R}$ is defined for $\gamma \in \Gamma$ by

$$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$

for a fixed $x_0 \in X$. The "Quantitative Bounded Distance Theorem" provides a uniform comparison between the orbit distance and the stable norm:

$$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$

where $D = \mathrm{diam}(\Gamma \backslash X)$ and $$\omega = \omega(\Gamma, d) = \lim_{R \to \infty} \#\{\gamma : d(x_0, \gamma.x_0) < R\}/R^n$$ (asymptotic volume). An explicit, scale-invariant estimate is:

$(X, d)$0

This result establishes near-isometry between $(X, d)$1 and $(X, d)$2 at the macroscopic scale, with constants depending only on the dimension, co-diameter, and growth.

Further, a refined "Abelian Margulis Lemma" gives optimal two-sided bounds for the stable systole:

$(X, d)$3

sharp in both lower and upper bounds. Each parameter $(X, d)$4, $(X, d)$5, and $(X, d)$6 is provably necessary: examples show that omitting any can make the defect $(X, d)$7 arbitrarily large (Cerocchi et al., 2014).

2. Quantitative Estimates in Optimal Transport and Probability

Explicit control over Wasserstein distances underlies quantitative sensitivity analysis for PDEs and SDEs. For Fokker-Planck flows $(X, d)$8 and $(X, d)$9 parametrized by $\|\cdot\|_{st}$0, and under uniform ellipticity and Lipschitz constants $\|\cdot\|_{st}$1:

$\|\cdot\|_{st}$2

for $\|\cdot\|_{st}$3, with explicitly computable $\|\cdot\|_{st}$4.

For highly regularized drift-diffusions such as the overdamped Langevin process,

$\|\cdot\|_{st}$5

with $\|\cdot\|_{st}$6, and $\|\cdot\|_{st}$7 computable from model parameters. Sensitivity is Lipschitz in the parameters, with explicit exponential decay or growth depending on convexity (Morange, 3 Feb 2026).

In multivariate central limit theorems, explicit Wasserstein distance bounds are given for martingales and M-estimators. For a mean-zero martingale difference sequence with finite third moments,

$\|\cdot\|_{st}$8

for any $\|\cdot\|_{st}$9, yielding rates $\Gamma \otimes \mathbb{R}$0 under uniform moment bounds (Röllin, 2017).

Similarly, for general $\Gamma \otimes \mathbb{R}$1-estimators in parametric models, provided suitable concentration and regularity conditions:

$\Gamma \otimes \mathbb{R}$2

with explicit dependence of $\Gamma \otimes \mathbb{R}$3 on complexity and moment constants. The leading rate is minimax-optimal up to logarithmic factors (Bachoc et al., 2021).

3. Quantitative Bounds for Distance Problems in Discrete Geometry and Additive Combinatorics

For $\Gamma \otimes \mathbb{R}$4-distance sets $\Gamma \otimes \mathbb{R}$5 contained in a box $\Gamma \otimes \mathbb{R}$6, $\Gamma \otimes \mathbb{R}$7, the slice-rank polynomial method yields

$\Gamma \otimes \mathbb{R}$8

where $\Gamma \otimes \mathbb{R}$9. This upper bound is essentially optimal for large $\gamma \in \Gamma$0 (Hegedüs, 2018).

In finite field settings, for $\gamma \in \Gamma$1, $\gamma \in \Gamma$2, and the algebraic distance set $\gamma \in \Gamma$3,

$\gamma \in \Gamma$4

and if $\gamma \in \Gamma$5, then

$\gamma \in \Gamma$6

The improvement over previous exponents arises from refined control of rectangle and isosceles triangle counts in $\gamma \in \Gamma$7 (Iosevich et al., 2019).

Sharp quantitative Ramsey-type bounds are obtained for the distance Ramsey number $\gamma \in \Gamma$8:

$\gamma \in \Gamma$9

for fixed $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$0 and large $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$1, bridging geometric extremal combinatorics with classical Ramsey theory (Kupavskii et al., 2013).

4. Quantitative Approximation in High-Dimensional Geometry and Probability Metrics

Hausdorff and Gromov-Hausdorff distances admit explicit multi-dimensional control:

• For the Gromov-Hausdorff distance $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$2 between unit $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$3- and $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$4-dimensional spheres (w.r.t. geodesic metric),

$$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$5

and, for $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$6, $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$7, realized via explicit antipodal Voronoi tessellations (Harrison et al., 2023).

• For approximations of $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$8-distance functions (e.g., empirical geometric medians) in high dimensions, a set of $$\|\gamma\|_{st} := \lim_{k \to \infty} \frac{1}{k} d(x_0, \gamma^k.x_0)$$9 random points on $x_0 \in X$0 yields a rescaled halving polyhedron within $x_0 \in X$1 of the unit ball, with high probability. Complexity lower bounds for $x_0 \in X$2-approximations by distance-like functions are exponential in $x_0 \in X$3 (Mérigot, 2013).
• For Zolotarev distances $x_0 \in X$4 and their one-dimensional projections, the quantitative Cramér–Wold theorem gives

$x_0 \in X$5

with $x_0 \in X$6. In the compact-support regime this simplifies to a rate with $x_0 \in X$7, known to be sharp (Bobkov et al., 21 Jun 2025).

5. Quantum Information: Quantitative Bounds for Distinguishability, Resources, and Statistical Distance

Explicit degree-two polynomial bounds are central to quantum average-case distance measures between states, measurements, and channels. For $x_0 \in X$8-dimensional systems and $x_0 \in X$9 a $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$0-approximate 4-design,

• States: $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$1
• Measurements: $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$2
• Channels: $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$3

These metrics enable tight operational lower bounds via random-circuit measurements:

• Worst-case:average-case separation obeys $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$4 for states, $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$5 for measurements, $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$6 for channels (Maciejewski et al., 2021).

For geometric quantum resource quantification,

$$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$7

where $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$8 is the sum of diamond norms between measure-and-prepare channels determined by the POVMs (Tendick et al., 2022). Analytical incompatibility bounds for rank-1 projectors and MUB assemblages are tight and saturate $$|d(x, \gamma.x) - \|\gamma\|_{st}| \leq C(n, D, \omega),$$9 for specific families.

6. Sharp Quantitative Information-Theoretic Bounds

Correlation distance $D = \mathrm{diam}(\Gamma \backslash X)$0 (classical) and $D = \mathrm{diam}(\Gamma \backslash X)$1 (quantum) control mutual information via dimension-dependent inequalities sharper than Pinsker's:

$D = \mathrm{diam}(\Gamma \backslash X)$2

with quantum analogues $D = \mathrm{diam}(\Gamma \backslash X)$3 providing lower bounds for $D = \mathrm{diam}(\Gamma \backslash X)$4 (Hall, 2013). These results yield operational criteria for entanglement and strong constraints in quantum simulation and Bell nonlocality.

7. Algorithmic and Fractal Aspects: Algorithmic Information and Hausdorff Dimension Bounds

Quantitative algorithmic information bounds on distances and projections ensure that for $D = \mathrm{diam}(\Gamma \backslash X)$5, under effective dimension independence,

$D = \mathrm{diam}(\Gamma \backslash X)$6

where $D = \mathrm{diam}(\Gamma \backslash X)$7 denotes finite-precision Kolmogorov complexity up to $D = \mathrm{diam}(\Gamma \backslash X)$8 bits. These imply, via the point-to-set principle, explicit lower bounds on the Hausdorff dimensions of pinned distance sets:

$D = \mathrm{diam}(\Gamma \backslash X)$9

for any analytic $$\omega = \omega(\Gamma, d) = \lim_{R \to \infty} \#\{\gamma : d(x_0, \gamma.x_0) < R\}/R^n$$0, $$\omega = \omega(\Gamma, d) = \lim_{R \to \infty} \#\{\gamma : d(x_0, \gamma.x_0) < R\}/R^n$$1 (Cholak et al., 5 Sep 2025). The structure of exceptional sets for projections and distances is characterized precisely in terms of optimal Hausdorff oracles.

---

Quantitative distance bounds thus form a rigorous backbone for stability, estimation, complexity, and resource quantification across geometric, probabilistic, combinatorial, and quantum domains. Such results not only supply universal comparison scales but also guide sharpness, parametric dependence, and the identification of structural obstructions or necessary conditions.