McMullen's 1998 paper "Lipschitz maps and nets in Euclidean space" connected three questions about the Euclidean space Rd\mathbb{R}^d of dimension d≥2d\ge 2: whether every positive bounded density is the Jacobian determinant of a bi-Lipschitz homeomorphism; which bounded functions are divergences of Lipschitz or quasiconformal vector fields; and Gromov's 1993 question whether every separated net is bi-Lipschitz equivalent to the integer lattice Zd\mathbb{Z}^d. All three admit negative endpoint answers — the first and third independently by Burago and Kleiner, the second via Ornstein's L1L^1 non-inequality and independently by Preiss — while one step below Lipschitz all three hold universally: the Rivière–Ye prescribed volume form theorem realizes every density bounded above and below by a bi-Hölder homeomorphism, every bounded datum has a divergence with Zygmund-class components, and any two separated nets are related by a homogeneous bi-Hölder bijection. The open problem, now shaped by two explicit published conjectures, is the exact regularity threshold between these endpoints on the scale of moduli of continuity ω\omega with t≪ω(t)≪tβt\ll\omega(t)\ll t^\beta for every β∈(0,1)\beta\in(0,1), such as ω(t)=tlog⁡(1/t)p\omega(t)=t\log(1/t)^p. Dymond and Kaluža constructed separated nets and continuous densities that are irregular at ω(t)=tlog⁡(1/t)α0\omega(t)=t\log(1/t)^{\alpha_0} for some α0(d)\alpha_0(d), which may be taken in (0,1)(0,1) — indeed arbitrarily small for the density construction — and conjectured that irregular nets exist exactly for the moduli asymptotically smaller than every Hölder modulus; Inoquio-Renteria and Viera proved positive rectifiability and solvability results for large intermediate moduli and state as a plausible conjecture that bi-ω\omega solutions exist whenever tlog⁡(1/t)≲ω(t)t\log(1/t)\lesssim\omega(t). The surviving open core is the sharp threshold for densities and nets: both directions of the Dymond–Kaluža conjecture and the equivalence between the density and net statements at intermediate moduli.

Background

A classical theorem of Moser (1965) states that any two smooth positive volume forms of equal total volume on a compact manifold are related by a diffeomorphism; this initiated the prescribed Jacobian problem, which asks, for a given density ff (a positive measurable function), for a map ϕ\phi whose derivative has determinant ff almost everywhere. McMullen's 1998 paper studied this problem at the bi-Lipschitz level: a homeomorphism ϕ\phi is bi-Lipschitz if it distorts all distances by at most a fixed factor, i.e. 1K∣x−y∣≤∣ϕ(x)−ϕ(y)∣≤K∣x−y∣\frac{1}{K}|x-y|\le|\phi(x)-\phi(y)|\le K|x-y| for some constant KK and all x≠yx\neq y. The paper framed three questions for Rd\mathbb{R}^d, d≥2d\ge 2, linked by a common linearized core. First, the prescribed Jacobian question in density form: given f∈L∞f\in L^\infty (the class of measurable functions that are essentially bounded) with inf⁡f>0\inf f>0 and mean one, is there a bi-Lipschitz homeomorphism ϕ\phi with det⁡Dϕ=f\det D\phi=f almost everywhere? On the unit cube [0,1]d[0,1]^d, with the map fixing the boundary, this is equivalent to the volume-form equation

vol(Φ(E))=∫Ef dxfor every open E⊆[0,1]d.\mathrm{vol}(\Phi(E))=\int_E f\,dx\quad\text{for every open }E\subseteq[0,1]^d.

Second, the divergence question: given f∈L∞f\in L^\infty of mean zero on the torus Td=(R/Z)d\mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d (the unit cube with periodic boundary conditions), does div⁡v=f\operatorname{div}v=f admit a Lipschitz solution, or at least a quasiconformal one? Here a vector field vv is called quasiconformal when its conformal strain — the symmetric trace-free part of the derivative matrix, Sv=12(Dv+DvT)−1d(div⁡v)ISv=\frac{1}{2}(Dv+Dv^{T})-\frac{1}{d}(\operatorname{div}v)I — lies in L∞L^\infty; Lipschitz fields are exactly those with Dv∈L∞Dv\in L^\infty, so this relaxes the full derivative bound to its trace-free symmetric part, which is also the natural class for infinitesimal deformations generated by isotopies of bi-Lipschitz maps. Third, the net question, posed by Gromov in 1993: is every separated net — a discrete set Y⊂RdY\subset\mathbb{R}^d whose points are pairwise at distance at least ϵ>0\epsilon>0 while every point of space lies within distance R<∞R<\infty of YY, also called a Delone set — bi-Lipschitz equivalent to the integer lattice Zd\mathbb{Z}^d?

All three questions have negative answers for d>1d>1. For the divergence equation, McMullen deduced this from a 1962 theorem of Ornstein on differential operators: if P0,…,PmP_0,\dots,P_m are linearly independent constant-coefficient linear differential operators of the same order, then ∥P0g∥L1\|P_0g\|_{L^1} cannot be bounded by a constant times ∑i∥Pig∥L1\sum_i\|P_ig\|_{L^1}; applied to the Laplacian Δ\Delta versus the trace-free second-order operators, this produces f∈L∞f\in L^\infty that is not the divergence of any Lipschitz or quasiconformal field. Preiss independently proved non-solvability, by showing that Lipschitz solutions of PDE force additional regularity of the datum. For the density and net questions the counterexamples are multiscale: one constructs a density, bounded above and below, that oscillates on a sequence of cubes whose scale ratios grow sufficiently fast, so that any bi-Lipschitz realization would violate perimeter-to-area ratios at one scale using the slack granted at another. Burago and Kleiner independently constructed such non-realizable densities — theirs can be taken continuous, answering a question reportedly posed by Moser and Reimann in the 1960s — and McMullen proved the density and net statements equivalent at the bi-Lipschitz level, using Voronoi cell partitions (the partition of space into the regions nearest to each net point) and matching arguments based on Hall's marriage theorem (the classical criterion for the existence of an injective choice function) together with a Schröder–Bernstein argument.

In the opposite, softer direction, all three questions hold universally one step below Lipschitz. The linear theory of the Laplacian, via Calderón–Zygmund singular integrals (the Fourier-analytic technique controlling derivatives of a function by the function itself), gives, for every mean-zero f∈L∞(Td)f\in L^\infty(\mathbb{T}^d) — and on Rd\mathbb{R}^d for every f∈L∞f\in L^\infty — a solution of div⁡v=f\operatorname{div}v=f whose components lie in the Zygmund class — the functions satisfying ∣v(x+y)+v(x−y)−2v(x)∣≤C∣y∣|v(x+y)+v(x-y)-2v(x)|\le C|y| — a class just below Lipschitz. Rivière and Ye proved that the volume-form equation above admits, for every L∞L^\infty density bounded above and below, a bi-Hölder solution, where a map is Hölder with exponent α∈(0,1)\alpha\in(0,1) if ∣ϕ(x)−ϕ(y)∣≤K∣x−y∣α|\phi(x)-\phi(y)|\le K|x-y|^{\alpha}, and bi-Hölder if both ϕ\phi and ϕ−1\phi^{-1} are Hölder; the exponent α\alpha and constant KK depend only on the sum ∥f∥∞+∥1/f∥∞\|f\|_\infty+\|1/f\|_\infty of the upper bound on ff and on its reciprocal. Transporting this along Voronoi cells, McMullen showed that any two separated nets are related by a homogeneous bi-Hölder bijection, homogeneous meaning ∣ϕ(x)−ϕ(y)∣≤K R1−α∣x−y∣α|\phi(x)-\phi(y)|\le K\,R^{1-\alpha}|x-y|^{\alpha} whenever ∣x∣,∣y∣≤R|x|,|y|\le R: a scale-invariant form of the Hölder condition suited to unbounded sets. Thus the trio fails at the Lipschitz/quasiconformal endpoint and holds universally in the Hölder/Zygmund category, and the paper explicitly left the entire intermediate range undetermined.

The intermediate range is measured by moduli of continuity: concave increasing functions ω:(0,1]→(0,∞)\omega:(0,1]\to(0,\infty) with ω(t)→0\omega(t)\to 0 as t→0+t\to 0^{+} lying strictly between the Lipschitz modulus ω(t)=t\omega(t)=t and the Hölder moduli ω(t)=tα\omega(t)=t^{\alpha}, the standard family being ωp(t)=tlog⁡(1/t)p\omega_p(t)=t\log(1/t)^{p}. A map is ω\omega-regular if ∣ϕ(x)−ϕ(y)∣≤K ω(∣x−y∣)|\phi(x)-\phi(y)|\le K\,\omega(|x-y|), bi-ω\omega if both it and its inverse are, and for nets the homogeneous version ∣ϕ(x)−ϕ(y)∣≤K R ω(∣x−y∣/R)|\phi(x)-\phi(y)|\le K\,R\,\omega(|x-y|/R) for ∣x∣,∣y∣≤R|x|,|y|\le R is used; a net is ω\omega-rectifiable if it is boundedly equivalent to Zd\mathbb{Z}^d by such a map. Dymond and Kaluža proved that for each d≥2d\ge 2 there is α0(d)\alpha_0(d) — recordable in (0,1)(0,1), and admissible arbitrarily small for the density construction — and a separated net that is irregular already at ω(t)=tlog⁡(1/t)α0\omega(t)=t\log(1/t)^{\alpha_0}, together with a continuous density admitting no bi-ω\omega solution of the volume-form equation at that modulus, and that the realizable densities form a σ\sigma-porous (topologically small: a countable union of sets whose every point sees holes of arbitrarily large relative scale) subset of C([0,1]d)C([0,1]^d) and of L∞L^\infty; they conjectured the sharp threshold that irregular nets exist exactly for the moduli asymptotically smaller than every Hölder modulus, emphasizing that both implications are open. On the positive side, Inoquio-Renteria and Viera proved that every Delone set whose density deviation decays at rate O(1/i)O(1/i) across dyadic scales is ω\omega-rectifiable for every modulus in an explicit summability class containing all Hölder moduli and ωp\omega_p for p>dp>d; that ωp\omega_p-repetitive sets — sets in which every local pattern recurs, up to translation, within radius ≍r(log⁡r)p\asymp r(\log r)^{p} of any center, repetitivity being the uniform-recurrence condition familiar from aperiodic tiling theory — are rectifiable for p<1/dp<1/d; and that the volume-form equation admits bi-ω\omega solutions under an explicit oscillation-control condition satisfied by all Hölder moduli and by ωp\omega_p for p>1p>1. They state as a plausible conjecture — a conjectured sufficient condition, not an if-and-only-if claim — that bi-ω\omega solutions exist whenever tlog⁡(1/t)≲ω(t)t\log(1/t)\lesssim\omega(t). Since the proven failure moduli tlog⁡(1/t)α0t\log(1/t)^{\alpha_0} with α0<1\alpha_0<1 lie strictly below this conjectured positivity barrier, the two barriers bracket a nonempty open gap in which the exact threshold remains open.

Problem Statement

Fix d≥2d\ge 2. Let ω\omega range over concave moduli of continuity with t≪ω(t)≪tβt\ll\omega(t)\ll t^{\beta} as t→0t\to 0 for every β∈(0,1)\beta\in(0,1) — that is, moduli asymptotically above Lipschitz and below every Hölder modulus; the standard family is ωp(t)=tlog⁡(1/t)p\omega_p(t)=t\log(1/t)^{p}. Determine the sharp regularity threshold on this scale for McMullen's density and net questions.

(a) Densities. For exactly which moduli ω\omega does the volume-form equation

vol(Φ(E))=∫Ef dxfor every open E⊆[0,1]d,Φ∣∂[0,1]d=id,\mathrm{vol}(\Phi(E))=\int_E f\,dx\quad\text{for every open }E\subseteq[0,1]^d,\qquad \Phi|_{\partial[0,1]^d}=\mathrm{id},

admit a bi-ω\omega solution for every f∈L∞f\in L^\infty with inf⁡f>0\inf f>0 and total mass ∫[0,1]df dx=1\int_{[0,1]^d}f\,dx=1 (the boundary condition forces the mass)? The same question is asked on all of Rd\mathbb{R}^d, where no normalization is needed, for every f∈L∞(Rd)f\in L^\infty(\mathbb{R}^d) with inf⁡f>0\inf f>0: for exactly which of the same moduli ω\omega does the corresponding equation admit a bi-ω\omega solution? These are two instances with different boundary and normalization conditions, and it is not known whether they share a solvable modulus set; both must be settled for facet (a) to be resolved. Universality holds for every Hölder modulus (Rivière–Ye, transported by McMullen) and for the moduli satisfying the oscillation condition of Inoquio-Renteria–Viera (which includes ωp\omega_p for p>1p>1); it fails at ω=ωα0\omega=\omega_{\alpha_0} for α0(d)∈(0,1)\alpha_0(d)\in(0,1) arbitrarily small, even for continuous ff (Dymond–Kaluža). Inoquio-Renteria–Viera state as a plausible conjecture — a conjectured sufficient condition, not an if-and-only-if claim — that bi-ω\omega solutions exist whenever tlog⁡(1/t)≲ω(t)t\log(1/t)\lesssim\omega(t).

(b) Nets. For exactly which moduli ω\omega is every separated net X⊂RdX\subset\mathbb{R}^d related to Zd\mathbb{Z}^d by a homogeneous bi-ω\omega bijection — a bijection ff with ∣f(x)−f(y)∣≤K R ω(∣x−y∣/R)|f(x)-f(y)|\le K\,R\,\omega(|x-y|/R) whenever ∣x∣,∣y∣≤R|x|,|y|\le R, and the same for f−1f^{-1}? The Dymond–Kaluža conjecture predicts that irregular nets exist exactly when lim⁡t→0ω(t)/tβ=0\lim_{t\to 0}\omega(t)/t^{\beta}=0 for every β∈(0,1)\beta\in(0,1); both directions are open, as is the extension to intermediate moduli of McMullen's bi-Lipschitz-level equivalence between facets (a) and (b). Note that the Dymond–Kaluža net conjecture and the Inoquio-Renteria–Viera density conjecture are jointly consistent over moduli with tlog⁡(1/t)≲ω(t)≪tβt\log(1/t)\lesssim\omega(t)\ll t^{\beta} only if this equivalence fails at intermediate moduli. A sub-facet left explicitly open by Inoquio-Renteria–Viera: whether every ωp\omega_p-repetitive Delone set is ωp\omega_p-rectifiable for 1/d<p≤d1/d<p\le d.

The common objective is to locate exactly where, on the regularity scale between the Lipschitz endpoint (where McMullen's density and net questions fail) and the Hölder category (where they hold universally), the transition from counterexamples to universal solvability takes place, and to describe the solvable classes exactly. Facets (a) and (b) pose this for the density and net questions respectively, and are linked by the common threshold question. Completion standard: settling the exact solvable modulus set in at least one of the two facets resolves the task, since each settles the sharp threshold on this scale; for the density facet (a) this requires both the boundary-fixed cube instance and the unnormalized Rd\mathbb{R}^d instance. The extension of McMullen's density/net equivalence to intermediate moduli and the ωp\omega_p-repetitive sub-facet for 1/d<p≤d1/d<p\le d are related sub-goals: they remain open and are not required in addition to a resolved facet.

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