Classification of generator-remainder behaviours for meromorphic continuation of the renewal transform in Koch-snowflake fractal sprays

The Dirichlet eigenvalue counting function of the lattice fractal sprays $\Omega(k_1,k_2)\subset\mathbb{R}^2$ built from the Koch snowflake admits, by Kombrink–Schmidt (2023, published 2025), an asymptotic expansion whose poles lie strictly left of the critical line $\Re(z)=-\delta/2$, where $\delta=\log 4/\log 3$ is the boundary dimension, with remainder $o(t^{\delta/2+\gamma})$ for every $\gamma>0$. Whether the two-sided Fourier–Laplace transform driving the renewal analysis extends meromorphically across that line, and whether $z=-\delta/2$ is a genuine pole — heuristically, whether the counting function has a second term of order exactly $t^{\delta/2}$ — is undetermined: boundedness of the generator-level remainder does not suffice, since admissible oscillatory remainder components create lacunary series with dense singularities on the line. The open problem asks for a classification of the fine-scale generator-remainder behaviours (including the arithmetic constraints coming from the lattice constant $a=(\log 3)/2$) that are necessary and sufficient for meromorphic continuation of the transform and for the presence of a genuine pole at $-\delta/2$, together with the corresponding sharp second-term asymptotics for the counting function.

Background

For a bounded planar domain $\Omega$, let $N_D(\Omega,t)$ denote the Dirichlet eigenvalue counting function: the number of eigenvalues (counted with multiplicity) of $-\Delta$ on $\Omega$ with zero boundary conditions that do not exceed $t$. Weyl's law gives $N_D(\Omega,t)=\frac{\mathrm{vol}_2(\Omega)}{4\pi}t+o(t)$, and for smooth boundary a second term of order $\sqrt{t}$ proportional to the boundary length. Berry conjectured (the Weyl–Berry conjecture) that for rough boundaries the second term is governed by the fractal geometry of $\partial\Omega$; Lapidus's modified Weyl–Berry conjecture predicts, more precisely, that if $\partial\Omega$ has Minkowski dimension $d$ — the decay exponent of the area of the inner $\varepsilon$-neighbourhood $\Omega^{-\varepsilon}:=\{x\in\Omega:\mathrm{dist}(x,\partial\Omega)<\varepsilon\}$ as $\varepsilon\to0$ — and positive finite Minkowski content (the limit of $\varepsilon^{-(2-d)}\mathrm{vol}_2(\Omega^{-\varepsilon})$), then a second term of order $t^{d/2}$ should appear. Lapidus proved an early partial resolution, Lapidus–Pomerance established the one-dimensional case (fractal strings) exactly, and the same authors showed the modified conjecture is false in dimensions $\ge 2$; for planar domains with genuinely fractal boundary, the existence and shape of a sharp second term remains open in essentially all nontrivial cases.

A fractal spray is a countable disjoint union of rescaled copies of a fixed open set $G$, called the generator, the scalings being those of an iterated function system (IFS), i.e. a finite family of contracting similarities. If all contraction ratios are integer powers of a common $e^{-a}$, the IFS is called lattice, with lattice constant $a$. Kombrink and Schmidt study lattice sprays $\Omega(k_1,k_2)\subset\mathbb{R}^2$ with $(k_1,k_2)\in\{0,\dots,6\}^2$, built from the Koch snowflake $K$ of base length 1, whose boundary has Hausdorff and Minkowski dimension $\delta=\log 4/\log 3\approx 1.26$. The base case $\Omega(0,0)$ arises via a 12-map IFS with six maps of ratio $1/3=e^{-2a}$ and six of ratio $3^{-1/2}=e^{-a}$, so $a=(\log 3)/2$, and with generator $K$; for $(k_1,k_2)\neq(0,0)$ one replaces $k_1$ of the ratio-$1/3$ maps by six maps of ratio $1/9$ each and $k_2$ of the ratio-$3^{-1/2}$ maps by six maps of ratio $1/(9\sqrt{3})$ each, obtaining an IFS with $12+5(k_1+k_2)$ maps whose generator is a union of $1+k_1+k_2$ Koch snowflakes of different base lengths, all similar to $K$. Since $N_D$ is additive over disjoint unions and scales as $N_D(\alpha G,t)=N_D(G,\alpha^2 t)$, the counting function of each generator reduces to that of $K$, and for every $(k_1,k_2)$ the spray's counting function is a renewal-type sum of shifted copies of the generator counting function $g(t):=N_D(K,e^t)$. A companion result of the authors gives two-sided bounds on the generator remainder: $g(t)=\frac{\mathrm{vol}_2(K)}{4\pi}e^t+M(t)e^{t\delta/2}$ with $M$ bounded (explicit constants $C^-=-1481$, $C^+=281.5\cdot 10^3$ from Dirichlet–Neumann bracketing and inner Minkowski-content estimates).

The analysis proceeds through the two-sided Fourier–Laplace transform of the counting function sampled on the lattice $2a\mathbb{Z}+\beta$ (with $\beta\in[0,2a)$ a phase parameter):

$\hat N_\beta(z):=\sum_{\ell\in\mathbb{Z}} e^{2a\ell z}\, N_D\bigl(\Omega,e^{2a\ell+\beta}\bigr),$

which converges for $\Re(z)<-1$ and, on the strength of the boundedness of $M$, extends meromorphically to the half-plane $\mathbb{C}_{-\delta/2}=\{z:\Re(z)<-\delta/2\}$, with poles among the complex dimensions $Z_C=\{z:\sum_{i} r_i^{-2z}=1,\ \Im(z)\in[0,\pi/a)\}$ determined by the IFS ratios $r_i$. Transferring these poles back to the counting function yields their Theorem 4.1: for every $\gamma>0$,

$N_D\bigl(\Omega(k_1,k_2),e^t\bigr)=\tfrac{\mathrm{vol}_2(\Omega(k_1,k_2))}{4\pi}e^t-\sum_{z\in Z_C,\ \Re(z)<-\delta/2}\bigl[2a\, Q_{\beta(t)}(z)\bigr]\, e^{-2a\lfloor t/2a\rfloor z}+o\bigl(e^{t(\delta/2+\gamma)}\bigr),$

with coefficients bounded and oscillatory in $\beta(t)=2a\{t/2a\}$. All poles strictly left of the critical line $\Re(z)=-\delta/2$ are captured; the method stops at that line because the summand involving $M$ is only controlled by the geometric factor $(1-e^{2a(z+\delta/2)})^{-1}$, and whether $z=-\delta/2$ is a genuine pole — which would correspond to a second term of order exactly $t^{\delta/2}$ in $N_D$ — depends on the fine behaviour of $M$ that the two-sided bounds do not determine. Notably, $-\delta/2\notin Z_C$ (one checks $\sum_i r_i^{\delta}=6\cdot(1/3)^{\delta}+6\cdot 3^{-\delta/2}=9/2\ne 1$), so a pole at $-\delta/2$ could only originate from the generator remainder itself, making the question genuinely spectral rather than geometric. By contrast, the companion problem for the inner $\varepsilon$-parallel volume of the same sprays is completely settled in the same paper, including the critical-line pole at $2-\log 4/\log 3$, because the generator's tube function is explicitly known from the Lapidus–Pearse tube formula for the Koch snowflake curve.

Problem Statement

For the lattice Koch-snowflake sprays $\Omega(k_1,k_2)\subset\mathbb{R}^2$ of Kombrink–Schmidt ($(k_1,k_2)\in\{0,\dots,6\}^2$; built from Koch snowflakes similar to the snowflake $K$ of base length 1, boundary dimension $\delta=\log 4/\log 3$, lattice constant $a=(\log 3)/2$), with generator Dirichlet counting function $g(t)=N_D(K,e^t)=\frac{\mathrm{vol}_2(K)}{4\pi}e^t+M(t)e^{t\delta/2}$ and bounded remainder $M$, classify the fine-scale behaviour of $M$ — and of any additional oscillatory components of $g$ that are admissible, i.e. consistent with the two-sided bounds established for the generator remainder (Theorem 3.2: $M$ bounded, with the explicit constants $C^-=-1481$, $C^+=281.5\cdot 10^3$; the heuristic component constructed in Remark 4.2 below lies in this class) — that determines the analytic structure of the two-sided Fourier–Laplace transform $\hat N_\beta(z)=\sum_{\ell\in\mathbb{Z}}e^{2a\ell z}N_D(\Omega(k_1,k_2),e^{2a\ell+\beta})$ on and beyond the critical line $\Re(z)=-\delta/2$, as follows:

(i) give necessary and sufficient conditions on the generator-level remainder under which $\hat N_\beta$ admits a meromorphic extension across the line $\Re(z)=-\delta/2$, for every phase $\beta\in[0,2a)$ (extendability propagates across phases, so this quantification is effectively uniform in $\beta$);

(ii) give conditions under which $z=-\delta/2$ is a genuine pole of that extension — heuristically, whether $N_D(\Omega(k_1,k_2),t)$ possesses a second asymptotic term of order exactly $t^{\delta/2}$ (with a multiplicatively periodic coefficient, i.e. periodic in $\log t$), or whether the remainder is of smaller order, or no meromorphic continuation exists at all;

(iii) derive, for each classified remainder behaviour, the corresponding sharp limits on the possible secondary exponents and coefficient growth in the expansion of $N_D(\Omega(k_1,k_2),t)$ relative to the currently known remainder $o(t^{\delta/2+\gamma})$ valid for every $\gamma>0$.

The classification must account for the arithmetic obstruction exhibited in Remark 4.2 of the source: an admissible oscillatory component $A(t)e^{t\delta/2}$ of $g$ with $A(t)=1/\lfloor t/a\rfloor$ when $\lfloor t/a\rfloor$ is a power of two and $A(t)=0$ otherwise generates in the sampled transform the lacunary series $\sum_{m\ge 0}\bigl(e^{a(z+\delta/2)}\bigr)^{2^m}/2^m$ — a power series in $e^{a(z+\delta/2)}$ whose nonzero terms occur only at the widely spaced exponents $2^m$ — and, were a meromorphic extension beyond $\Re(z)=-\delta/2$ to exist, some phase-shifted transform would also contain the coefficient-one series $\sum_{m\ge 0}\bigl(e^{a(z+\delta/2)}\bigr)^{2^m}$, which diverges at every $z=-\delta/2+2\pi i q/(a\,2^p)$ ($p\in\mathbb{Z}$, $q\in\mathbb{N}$) — a set dense on the line $\Re(z)=-\delta/2$ — so no meromorphic extension beyond the line, and in particular no pole at $-\delta/2$, can be concluded from boundedness of the generator remainder alone. The research target comprises (a) the general classification (i)–(iii) over admissible remainders, including the diophantine/lattice-arithmetic conditions arising from $a$ and the scaling ratio; (b) a resolution of the instance question for the actual Koch snowflake generator — does the true remainder $M$ of $N_D(K,\cdot)$ permit meromorphic continuation, and is $-\delta/2$ a genuine pole?

References

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