For a bounded open set Ω⊂R\Omega\subset\mathbb{R} with fractal boundary F=∂ΩF=\partial\Omega of Minkowski dimension D∈(0,1)D\in(0,1), the work of Lapidus–Pomerance and its extension by He–Lapidus and by Eichinger–Winter show that if the boundary is hh-Minkowski measurable for a gauge function hh that is regularly varying of index 1−D1-D at 00, then the eigenvalue counting function N(λ)N(\lambda) of the Dirichlet Laplacian on Ω\Omega has an exact asymptotic second term N(λ)=φ(λ)−c1,DM(h;F)f(λ)+o(f(λ))N(\lambda)=\varphi(\lambda)-c_{1,D}\mathcal{M}(h;F)f(\sqrt\lambda)+o(f(\sqrt\lambda)), where f(x)=x h(1/x)f(x)=x\,h(1/x). The converse — whether the existence of such a second term forces hh-Minkowski measurability — is known to fail at every dimension DD at which the Riemann zeta-function has a zero on the line ℜ(s)=D\Re(s)=D (He–Lapidus), while for the classical power gauge Lapidus–Maier proved the converse fails at D=1/2D=1/2 and holds for all other D∈(0,1)D\in(0,1) if and only if the Riemann hypothesis is true. The open problem is to establish the corresponding precise characterization of the converse for general regularly varying gauges, and to determine the sharpness of the regular-variation framework itself: either to construct a bounded open set whose spectrum admits such a second-order asymptotic while its tube volume admits no regularly varying gauge, or to prove that no such example exists.

Background

For a bounded open set Ω⊂Rd\Omega\subset\mathbb{R}^d (interpreted as a vibrating membrane fixed along its boundary), the eigenvalues λ1≤λ2≤…\lambda_1\le\lambda_2\le\dots of the Dirichlet Laplacian are the natural vibration frequencies. The eigenvalue counting function N(λ):=#{i:λi≤λ}N(\lambda):=\#\{i:\lambda_i\le\lambda\} obeys Weyl's law: N(λ)∼φ(λ)N(\lambda)\sim\varphi(\lambda) as λ→∞\lambda\to\infty, where the Weyl term is φ(λ)=(2π)−dωd∣Ω∣dλd/2\varphi(\lambda)=(2\pi)^{-d}\omega_d|\Omega|_d\lambda^{d/2}, with ωd\omega_d the volume of the dd-dimensional unit ball. For smooth boundaries a second additive term of order λ(d−1)/2\lambda^{(d-1)/2}, proportional to the boundary surface area, is known under genericity assumptions, so one can 'hear' both the volume and the boundary measure of a smooth drum. For fractal boundaries the picture changes: Berry conjectured a second term involving the Hausdorff measure of the boundary; Brossard and Carmona disproved this and proposed Minkowski content instead, leading to the modified Weyl–Berry conjecture formulated by Lapidus.

The Minkowski dimension and Minkowski content describe the scaling of the parallel volume (tube volume) VF(r):=∣Fr∩Ω∣dV_F(r):=|F_r\cap\Omega|_d, where FrF_r is the rr-neighborhood of the boundary. If VF(r)∼M rd−DV_F(r)\sim M\,r^{d-D} as r↘0r\searrow 0 with 0<M<∞0<M<\infty, the boundary FF is Minkowski measurable with content MM and Minkowski dimension DD. In dimension d=1d=1, Lapidus and Pomerance proved the modified Weyl–Berry conjecture: if FF is Minkowski measurable of dimension D∈(0,1)D\in(0,1) with content MD(F)\mathcal{M}^D(F), then N(λ)=φ(λ)−c1,DMD(F)λD/2+o(λD/2)N(\lambda)=\varphi(\lambda)-c_{1,D}\mathcal{M}^D(F)\lambda^{D/2}+o(\lambda^{D/2}), with c1,Dc_{1,D} an explicit constant involving the Riemann zeta-function ζ\zeta. Their proof passes through the fractal string L=(lj)j∈N\mathcal{L}=(l_j)_{j\in\mathbb{N}} — the sequence of lengths of the disjoint open intervals composing Ω\Omega, in nonincreasing order — and the packing defect δ(x)=∑j{ljx}\delta(x)=\sum_j\{l_jx\}, where {y}\{y\} denotes the fractional part of yy.

Many boundaries are not Minkowski measurable: VF(r)V_F(r) may degenerate (00 or ∞\infty content) or oscillate. He and Lapidus extended the theory to generalized Minkowski contents based on gauge functions h:(0,∞)→(0,∞)h:(0,\infty)\to(0,\infty) more general than powers: the hh-Minkowski content is M(h;F):=lim⁡r↘0VF(r)/h(r)\mathcal{M}(h;F):=\lim_{r\searrow 0}V_F(r)/h(r) when the limit exists in (0,∞)(0,\infty), and FF is then hh-Minkowski measurable. Their admissible gauge class G1−D\mathcal{G}_{1-D} carried technical conditions whose geometric meaning was unclear. Eichinger and Winter revisited this theory using Karamata theory (the analysis of regularly varying functions): a function hh is regularly varying at 00 with index ϱ\varrho (written h∈Rϱh\in\mathcal{R}_\varrho) if lim⁡y↘0h(ty)/h(y)=tϱ\lim_{y\searrow 0}h(ty)/h(y)=t^\varrho for every t>0t>0; SRϱ\mathcal{SR}_\varrho is the subclass of C1C^1 gauges. They proved the classes Gϱ\mathcal{G}_\varrho, Rϱ\mathcal{R}_\varrho, SRϱ\mathcal{SR}_\varrho are asymptotically equivalent, so regular variation is the essential hypothesis, and established a main theorem (Theorem 2.5 of the published version): for h∈SR1−Dh\in\mathcal{SR}_{1-D} with auxiliary functions f(x)=x h(1/x)f(x)=x\,h(1/x) and g=H−1(1/x)g=H^{-1}(1/x) where H(x)=x/h(x)H(x)=x/h(x), the hh-Minkowski measurability of FF is equivalent to lj∼L g(j)l_j\sim L\,g(j) for some L>0L>0, and implies the exact second-order spectral asymptotic N(λ)=φ(λ)−c1,DM(h;F)f(λ)+o(f(λ))N(\lambda)=\varphi(\lambda)-c_{1,D}\mathcal{M}(h;F)f(\sqrt\lambda)+o(f(\sqrt\lambda)) with c1,D=2D−1π−D(1−D)(−ζ(D))c_{1,D}=2^{D-1}\pi^{-D}(1-D)(-\zeta(D)).

The converse question — can the spectrum alone force the geometry? — has a number-theoretic twist. Lapidus and Maier showed for the power gauge h(r)=r1−Dh(r)=r^{1-D} that the converse fails at the midfractal dimension D=1/2D=1/2, and holds for every other D∈(0,1)D\in(0,1) if and only if the Riemann hypothesis is true, giving an inverse-spectral characterization of RH. He and Lapidus later proved (Theorem 2.9 of their Memoir) that for generalized gauges the converse fails at every DD for which ζ\zeta has a zero on the line ℜ(s)=D\Re(s)=D — a statement Eichinger and Winter note applies to all h∈SR1−Dh\in\mathcal{SR}_{1-D}. What is missing, and what Lapidus–Maier's theorem shows is achievable in the power case, is the complementary positive direction and a full characterization for general regularly varying gauges; Eichinger and Winter also record that it is unknown whether their regular-variation hypothesis is sharp, i.e. whether fractal strings with comparable spectral asymptotics can lie outside every regularly varying gauge class, and they express optimism that Karamata methods could simplify the known negative converse proofs. Related higher-dimensional bounds of He–Lapidus, and the question of whether the complex-dimensions (fractal zeta function) framework of Lapidus–Radunović–Žubrinić captures weakly degenerate sets — those whose tube volume degenerates at the power-law rate but is regularized by a non-power gauge — are further open directions recorded in the same source; the logarithmic-gauge case has since been partially incorporated into that framework through higher-order poles of the tube zeta function.

Problem Statement

Let Ω⊂R\Omega\subset\mathbb{R} be a bounded open set with boundary F=∂ΩF=\partial\Omega of Minkowski dimension D∈(0,1)D\in(0,1), let L=(lj)j∈N\mathcal{L}=(l_j)_{j\in\mathbb{N}} be the associated fractal string, and let N(λ)N(\lambda) be the eigenvalue counting function of the Dirichlet Laplacian on Ω\Omega, with Weyl term φ(λ)=∣Ω∣λ1/2/π\varphi(\lambda)=|\Omega|\lambda^{1/2}/\pi. For a gauge h∈SR1−Dh\in\mathcal{SR}_{1-D} (a C1C^1 gauge function, regularly varying at 00 with index 1−D1-D), set f(x)=x h(1/x)f(x)=x\,h(1/x). Eichinger–Winter (Theorem 2.5) proved that hh-Minkowski measurability of FF implies

N(λ)=φ(λ)−c1,D M(h;F) f(λ)+o(f(λ)),λ→∞,N(\lambda)=\varphi(\lambda)-c_{1,D}\,\mathcal{M}(h;F)\,f(\sqrt\lambda)+o\big(f(\sqrt\lambda)\big),\qquad\lambda\to\infty,

with c1,D=2D−1π−D(1−D)(−ζ(D))c_{1,D}=2^{D-1}\pi^{-D}(1-D)(-\zeta(D)). The problem consists of two linked components.

(1) Converse (inverse spectral) problem. Suppose that for some gauge h∈SR1−Dh\in\mathcal{SR}_{1-D} and some constant CC one has

N(λ)=φ(λ)−C f(λ)+o(f(λ)),λ→∞.N(\lambda)=\varphi(\lambda)-C\,f(\sqrt\lambda)+o\big(f(\sqrt\lambda)\big),\qquad\lambda\to\infty.

Does it follow that FF is hh-Minkowski measurable, necessarily with C=c1,D M(h;F)C=c_{1,D}\,\mathcal{M}(h;F)? It is known that the converse fails for every D∈(0,1)D\in(0,1) at which the Riemann zeta-function has a zero on the line ℜ(s)=D\Re(s)=D (He–Lapidus, Theorem 2.9, covering all such regularly varying gauges), while for the power gauge h(r)=r1−Dh(r)=r^{1-D} the full characterization is known: the converse fails at D=1/2D=1/2 and holds for all other D∈(0,1)D\in(0,1) if and only if the Riemann hypothesis is true (Lapidus–Maier). Establish the corresponding precise characterization for general regularly varying gauges: determine for which pairs (D,h)(D,h), h∈SR1−Dh\in\mathcal{SR}_{1-D}, the existence of the asymptotic second term implies hh-Minkowski measurability of the boundary, and for which pairs it fails.

(2) Sharpness of the regular-variation framework. Does there exist a bounded open set Ω⊂R\Omega\subset\mathbb{R} for which analogues of the second-order spectral asymptotics and of the equivalent geometric/fractal-string relations of Theorem 2.5 hold for some gauge function that is not regularly varying — in the source's words, a set "for which regularly varying gauge functions are not suitable but for which nevertheless results similar to Theorem 2.5 hold" — yet the volume function VF(r)=∣Fr∩Ω∣V_F(r)=|F_r\cap\Omega| is not asymptotic to M h(r)M\,h(r) for any regularly varying gauge h∈R1−Dh\in\mathcal{R}_{1-D} and any M∈(0,∞)M\in(0,\infty), i.e. an example genuinely outside regular variation? Either construct such an example or prove that none exists.

References

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