On finitely ramified (post-critically finite, p.c.f.) self-similar fractals such as the Sierpiński gasket, the eigenvalue counting function N(x)N(x) of the Laplacian satisfies a Weyl-type asymptotic N(x)=xdS/2(G(log⁡x/2)+o(1))N(x)=x^{d_S/2}(G(\log x/2)+o(1)) with a bounded periodic function GG, where dSd_S is the spectral dimension. For infinitely ramified self-similar sets (generalized Sierpiński carpets), Kajino (J. Funct. Anal. 2010) proved the analogous log-periodic asymptotics for the heat trace (spectral partition function) Zb(t)Z_b(t), and in the non-lattice case deduced Nb(x)∼c xdS/2N_b(x)\sim c\,x^{d_S/2} by Karamata's Tauberian theorem (a Tauberian theorem recovers the asymptotic behavior of a counting function from the asymptotic behavior of its Laplace–Stieltjes transform, here the partition function). However, in the lattice case — where the leading coefficient of Zb(t)Z_b(t) is a possibly non-constant periodic function — no asymptotics of the pointwise counting function Nb(x)N_b(x) are known, and Kajino explicitly leaves this as an open problem. The open core is: prove or disprove that, for self-similar Dirichlet forms on generalized Sierpiński carpets in the lattice case, Nb(x)=xdS/2(G^b(log⁡x/2)+o(1))N_b(x)=x^{d_S/2}(\hat{G}_b(\log x/2)+o(1)) for Dirichlet and Neumann boundary conditions with a bounded periodic G^b\hat{G}_b. The two obstructions identified in the source are that the required Tauberian conditions on the periodic principal part appear hopeless to verify when it is non-constant, and that no non-probabilistic control of the Neumann–Dirichlet difference NN(x)−ND(x)N_N(x)-N_D(x) is available to run a renewal argument.

Background

A generalized Sierpiński carpet is a self-similar compact set K⊂[0,1]dK\subset[0,1]^d obtained by repeatedly retaining a fixed pattern of subsquares (or subcubes) of the unit cube; it carries a natural self-similar Dirichlet form and associated Laplacian Δ\Delta (defined through resistance-form theory, i.e., a bilinear energy functional whose domain densely embeds in L2(K,μ)L^2(K,\mu), so that −Δ-\Delta is a non-negative self-adjoint operator with compact resolvent and hence a discrete spectrum 0≤λ1≤λ2≤⋯→∞0\le\lambda_1\le\lambda_2\le\cdots\to\infty). The eigenvalue counting function is Nb(x)=#{n:λn≤x}N_b(x)=\#\{n:\lambda_n\le x\}, with b∈{D,N}b\in\{D,N\} indicating Dirichlet or Neumann boundary conditions on the boundary of KK. The spectral dimension dSd_S is the exponent such that Nb(x)N_b(x) grows like xdS/2x^{d_S/2}, the fractal analogue of Weyl's law for the Laplacian on a bounded domain. On finitely ramified fractals (post-critically finite, p.c.f., sets — sets whose ramification locus is finite, so the set can be disconnected by removing finitely many points, the Sierpiński gasket being the prototype), Fukushima–Shima (1992) and Kigami–Lapidus (1993) established N(x)=xdS/2(G(log⁡x/2)+o(1))N(x)=x^{d_S/2}(G(\log x/2)+o(1)) where GG is a bounded nonnegative periodic function, using spectral decimation (an exact algebraic recursion expressing the eigenvalues of the fractal Laplacian through eigenvalues of finite discrete-graph approximations) and renewal arguments (a renewal equation is a functional equation of the form f(x)=∑if(γix)+g(x)f(x)=\sum_i f(\gamma_i x)+g(x), 0<γi<10<\gamma_i<1, whose solutions display periodic-in-log behavior when the γi\gamma_i are commensurably related — the lattice case). Generalized Sierpiński carpets are infinitely ramified: no spectral decimation is available, and the boundary contains infinitely many points at every scale, so the classical renewal mechanism, which needs the defect term NN(x)−ND(x)N_N(x)-N_D(x) to be bounded (as it is on p.c.f. sets, where the initial boundary graph is finite), cannot be applied. Kajino (2010) nonetheless proved, for self-similar Dirichlet spaces on carpets satisfying standard heat-kernel and self-similarity hypotheses, that the heat trace obeys Zb(t)=t−dS/2(G(−(log⁡t)/2)+o(1))Z_b(t)=t^{-d_S/2}(G(-(\log t)/2)+o(1)) as t↓0t\downarrow 0 with GG continuous and TT-periodic (Theorem 5.2), where Zb(t)=∑ne−λntZ_b(t)=\sum_n e^{-\lambda_n t} is the spectral partition function; in the non-lattice case Karamata's Tauberian theorem converts this into Nb(x)∼c xdS/2N_b(x)\sim c\,x^{d_S/2} (Corollary 5.4). The lattice case is the remaining gap: Kajino's Section 9 states that the periodic function GG of Theorem 5.2 may be non-constant, in which case it seems hopeless to verify the Tauberian conditions on GG needed to invert the transform, and that applying the renewal theorem directly to the counting function seems extremely difficult because the required estimate of NN(x)−ND(x)N_N(x)-N_D(x) cannot be obtained from probabilistic arguments. Hambly (2011) independently obtained the same partition-function asymptotics on carpets, together with periodic leading terms for the heat content (the total heat remaining in the set at time tt, ∫Kpt(x,x) dμ\int_K p_t(x,x)\,d\mu-type quantity obtained by integrating the heat kernel) and the integrated (cumulative) counting function, but pointedly not for the pointwise N(x)N(x); Kajino's 2014 follow-up strengthened the partition-function side to a full log-periodic asymptotic expansion, again without inverting to the counting function. Later work on counting-function asymptotics with periodic second terms (Alonso Ruiz–Freiberg 2017 on non-self-similar Hanoi attractors; Gu–Qiu 2025 on p.c.f. sets with strongly recurrent forms) has remained on the finitely ramified or otherwise decidable side of this boundary.

Problem Statement

For the self-similar Dirichlet form (Laplacian) on a generalized Sierpiński carpet satisfying the standard hypotheses of Kajino (J. Funct. Anal. 258 (2010) 1310–1360) — self-similarity of the form, uniform heat-kernel bounds, and the resulting spectral dimension dSd_S — settle the lattice case of the eigenvalue counting function asymptotics, which the source explicitly leaves open. Here the lattice case means that the spectral scaling factors γi∈(0,1)\gamma_i\in(0,1) of the self-similar Laplacian have commensurable logarithms: the additive subgroup ∑i∈SZlog⁡γi⊂R\sum_{i\in S}\mathbb{Z}\log\gamma_i\subset\mathbb{R} is discrete (equivalently, the ratios log⁡γi/log⁡γj\log\gamma_i/\log\gamma_j are rational); in this regime the partition function satisfies Zb(t)=t−dS/2(G(−(log⁡t)/2)+o(1))Z_b(t)=t^{-d_S/2}(G(-(\log t)/2)+o(1)) as t↓0t\downarrow 0 with a possibly non-constant TT-periodic continuous GG (Kajino, Theorem 5.2(2)). Determine the pointwise asymptotic behavior of Nb(x)=#{n:λn≤x}N_b(x)=\#\{n:\lambda_n\le x\} as x→∞x\to\infty, for both Dirichlet (b=Db=D) and Neumann (b=Nb=N) boundary conditions: prove or disprove that Nb(x)=xdS/2(G^b(log⁡x/2)+o(1))N_b(x)=x^{d_S/2}(\hat{G}_b(\log x/2)+o(1)) for some bounded periodic function G^b\hat{G}_b (in particular, whether log-periodic oscillation of x−dS/2Nb(x)x^{-d_S/2}N_b(x) occurs at all, as it does on finitely ramified p.c.f. fractals).

References

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