PRH | Essay | 7.21 • Fractal Dimension as a Mellin Slope under Positive Blur

Fractal Dimension as a Mellin Slope under Positive Blur A Blur–Native Framework with a Certified Overlap Case Study Aleksandar Perišić November 2025 Abstract We formalize a blur–native route to fractal dimensions. A positive, normalized blur (e.g. a Gaussian heat kernel) acts as a lawful switch between additive and multiplicative descriptions; exponents then emerge as log–log slopes of blurred energies. We present two complementary routes: an L2 –correlation route (for measures) and a Minkowski–perimeter route (for sets). Both turn dimension into a Mellin slope read directly off a monotone functional, coupled to a single legal switch at the end. Finally, we give a concrete overlapping, graph–directed self–similar example in one dimension where coverings are awkward, yet the blur method certifies the correlation dimension by solving a one–parameter spectral radius equation. 1 Blur as a lawful scaling probe Let Gε≥0be a family of positive,normalized kernels on Rdforming an approximate identity: Gε(x) = (4πε2)−d/2e−|x|2/(4ε2)(Gaussian/heat kernel), with RRdGε ( x ) dx = 1 and Gε∗Gε′ = G√ε2+ε′2 . Blurring by Gε is precisely the operational content of “blur”: we legalize a scale cut, keep positivity, and postpone any sharp limit to the end. The Fourier side reads c Gε(ξ)=e−ε2|ξ|2,∥µ∗Gε∥2 L2= (2π)−dZRd|b µ(ξ)|2e−2ε2|ξ|2dξ, so the blur acts as a log-Gaussian Mellin probe on frequencies. Reading a power–law slope in ε becomes a Tauberian statement under positivity: one clean readout, one switch, no illegal back-and-forth. Dimension as a slope (Mellin intuition) The representation r−s = csR∞ 0ts/2−1e−tr2dt exhibits Riesz energies as Mellin transforms of heat kernels. Thus, whenever a quantity scales as εα under heat blur, its exponent α is a Mellin slope. Dimension statements are therefore equivalent to the existence and identification of such slopes for positive blurred functionals. Constructive vs. ideal layers under blur From a blur–native perspective, it is useful to distinguish two kinds of points that typically live inside the same fractal. Consider a constructive scheme that produces approximants Fn at scales εn↓ 0(for instance, the n -th union of cylinders in an IFS construction, or a discretization at mesh size εn ). Define the stage layer Fstage := [ n≥0 Fn, 1 the set of points that appear at some finite stage and then never move again. In many familiar examples, points in Fstage can even be tagged by the first blur level εn at which they become uniquely determined (e.g. an endpoint produced at step nof a Cantor-like construction). The full fractal Fis obtained by idealizing: F:= Fstage, so that Ftypically decomposes as F=Fstage ⊔Fideal, Fideal := Fstage \Fstage. Points in Fideal are never present in any single finite approximant Fn ; they arise only as accumulation points of the stage layer, i.e. as the outcome of an infinite limiting process (blur ε→0). Thus, conceptually, there are: • stage points, created by explicit rules at some positive blur and stabilized thereafter, and •ideal points, created only by the blur–zero completion (closure). At any fixed positive blur ε > 0, this distinction disappears operationally: both stage and ideal points are indistinguishable at resolution ε , and they contribute in exactly the same way to blurred quantities such as the correlation functional C ( ε )or the perimeter functional M ( ε ). The Mellin–slope definitions of dimension below are deliberately formulated in this positive–blur world: we work only with monotone blurred functionals, and read a single exponent lim ε↓0 log C(ε) log εor lim ε↓0 log M(ε) log ε as εtends to zero. Conceptually, however, the blur viewpoint keeps track of provenance: stage points belong to the part of the fractal that is generated by lawful, finite–budget operations at positive blur, while the ideal layer Fideal is created only by invoking the blur–zero completion. The sharp fractal F is therefore best seen as the blur–invariant meaning of the entire family of blurred approximants ( Fn, εn )rather than as a configuration that first “appears” at ε = 0. The dimension theory below uses only the positive–blur data; the ideal layer is present precisely to the extent that it is forced by the completion of these blurred statistics. 2 Two blur–native routes to dimension 2.1 Route C: Correlation via L2blur (measures) Let µbe a finite Borel measure on Rd. Define the blurred correlation integral C(ε) := ZZRd×RdGε(x−y)dµ(x)dµ(y) = ∥µ∗Gε/√2∥2 L2(Rd).(1) Proposition 1 (Correlation dimension as blurred slope).Assume C ( ε ) ≍εD2−d as ε↓ 0. Then D2=d+ lim ε↓0 log C(ε) log ε. If µ is exact dimensional and D2 exists, then D2 agrees with the usual correlation dimension (and often with Hausdorff/Minkowski dimension in classical settings). Remark 2. Positivity of Gε ensures ε7→ C ( ε )is log-convex and monotone, aiding the existence and uniqueness of the slope. On the Fourier side, (1) is a positive Laplace transform in the parameter t= 2ε2, i.e. a textbook Tauberian situation. 2 2.2 Route M: Minkowski via blurred perimeter (sets) Let E⊂Rdbe measurable and uε:= 1E∗Gεits blurred indicator. Definition 3 (Blurred perimeter scale).Define the (total variation) functional M(ε) := ZRd|∇uε(x)|dx. Proposition 4 (Minkowski dimension as blurred slope).If M(ε)≍εd−1−DMas ε↓0, then DM=d−1−lim ε↓0 log M(ε) log ε, which coincides with the Minkowski (box) dimension in standard regularity regimes. Remark 5. M ( ε )measures the magnitude of a smoothed boundary layer. For fractal boundaries the scaling of this layer encodes the codimension. As with C , positivity yields monotonicity and log-convexity, stabilizing the slope. 3 A Blur–Dimension Protocol (P1) Pick a positive probe. Choose Gε≥ 0, RGε = 1 (Gaussian/Poisson/heat), avoiding oscillations. (P2) Select a monotone scalar. For measures, C(ε); for sets, M(ε). (P3) Work in the blurred world. Establish sub(add/mult)-multiplicative or renewal inequalities across scales for the blurred scalar (no sharp coverings). (P4) Read a single slope. Prove existence / uniqueness of lim ε↓0 log(·) log ε via positivity / logconvexity or Kingman / Fekete arguments. (P5) Switch once. Translate the exponent back to the intended dimension (Mellin/energy dictionary), with the switch performed after the monotone limit is certified. 4 Self-similar heuristics and quick wins Consider a self-similar measure µ with similarities Si ( x ) = rix + ti and weights pi ( Pipi = 1). Formally, µ=X i piµ◦S−1 i. Convolving with Gε and using Gε◦Si = r−d iGε/ri ( ·−ti )gives a multi-scale identity. Squaring and integrating yields, up to uniformly bounded multiplicative errors, C(ε)≈X i,j pipjr−d ir−d jC ε qr2 i+r2 j +lower-order terms.(2) A homogeneity ansatz C ( ε ) ≍εD2−d transforms (2) into a one-line balance that pins D2 . This recovers the classic values for Cantor/Sierpiński/Koch, now as a property of a positive blurred scalar rather than a sharp covering. 3 5 Case study: an overlapping graph-directed example certified by blur We now present a one-dimensional construction with genuine overlaps where covering arguments are fiddly, but the blur method gives a complete certification of the correlation dimension via a 2 × 2positive matrix. The example is deliberately simple yet nontrivial; it illustrates how blur replaces delicate transversality by a spectral radius computation. 5.1 The system Let G be a directed graph with vertices {A, B} and the following similarities on R with common ratio r= 1/3: Edges A→A:S1(x) = 1 3x, S2(x) = 1 3x+1 3, Edge A→B:S3(x) = 1 3x+2 3, Edges B→A:S4(x) = 1 3x+1 9, S5(x) = 1 3x+4 9. Assign probabilities p1 = p2 = α/ 2, p3 = 1 −α on A -outgoing edges (with state probability mass normalized at A ) and q4 = q5 = 1 / 2on B -outgoing edges. The system is graph-directed in the sense of Mauldin–Williams: words follow the graph, and the state of the cylinder is its terminal vertex. By construction the A -cylinders from S1 and S2 touch at their endpoints, while interactions with S3, S4, S5 create small exact overlaps across states; these make direct box coverings annoying. Let µA, µB be the limiting measures on the states (normalized so that the combined measure µ:= 1 2(µA+µB)is a probability). Define the blurred correlations CA(ε):=∥µA∗Gε∥2 L2,CB(ε):=∥µB∗Gε∥2 L2,C(ε) := 1 2(CA(ε)+CB(ε)). 5.2 A positive renewal system for CA,CB Using the self-similarity and positivity of Gε , cross-terms among distinct first-level cylinders are controlled by the blur (Gaussian tails), and one obtains a two-state renewal inequality valid for all sufficiently small ε: CA(ε) CB(ε)!≤r−dM(s) CA(ε/r) CB(ε/r)!+δ(ε), d = 1,(3) where s is a formal exponent (to be matched with the correlation dimension), δ ( ε )is a uniformly bounded error vector with δ(ε)=o(εβ)for some β > 0, and the 2×2positive matrix M(s)is M(s)=rs α2 2+(1−α)2α(1 −α) α/2 1/2!,i.e. M(s) = rs α2 2+(1−α)2α(1 −α) α/2 1/2!. Intuitively, entries count (with weights pipj ) the contributions from pairs of first-level cylinders landing in given states after one scale step; positivity of the blur converts potentially oscillatory interference into a monotone linear operator. Lemma 6 (Log-subadditivity & slope existence).Let d = 1 and r = 1 / 3. There exists ε0> 0 and C≥1such that for all nwith rnε0<1, C(ε0)≤C r−nM(s)nC(rnε0) + n−1 X k=0 C r−kM(s)kδ(rkε0), where C(ε) = (CA(ε),CB(ε))⊤. Consequently, the limit lim n→∞ 1 nlog C(rnε0) 4 exists and equals log r−log ρM(s)up to o(1), where ρ(·)denotes the spectral radius. Proof sketch. Iterate (3) ; positivity permits dropping cross-cancellations. Normalize by r−n and apply Gelfand’s formula to M( s ) n (Perron–Frobenius applies since the matrix is strictly positive when α∈ (0 , 1)). The error series stays bounded because δ ( rkε0 )decays with k by Gaussian tails. 5.3 Certification of the correlation dimension Set d = 1. The correlation ansatz C ( ε ) ≍εD2−1 translates under the one-step scaling ε7→ ε/r into the spectral balance rD2−1≈ρM(D2). The monotonicity in s of the RHS (matrix entries are rs times constants) and the strict log-convexity in εof Cguarantee a unique crossing. Theorem 7 (Certified D2 for the overlapping graph-directed example).For every α∈ (0 , 1) the correlation dimension D2of the measure µ=1 2(µA+µB)exists and is uniquely determined by rD2−1=ρ rD2 α2 2+(1−α)2α(1 −α) α/2 1/2!!=rD2ρ α2 2+(1−α)2α(1 −α) α/2 1/2!!.(4) Equivalently, D2= 1 + log ρα2 2+(1−α)2α(1−α) α/2 1/2 log(1/r)with r=1 3. Proof. By Lemma 6, the blurred correlations obey a positive linear renewal system across scales. Perron–Frobenius yields a single Lyapunov exponent; matching with the one-step rescaling of ε fixes the slope. Uniqueness follows from monotonicity in sand log-convexity of C(ε). Remark 8 (Explicit numbers).For the symmetric choice α=1 2, K:= α2 2+(1−α)2α(1 −α) α/2 1/2!α=1/2 = 1 8+1 4 1 4 1 4 1 2!= 3 8 1 4 1 4 1 2!. Its spectral radius is ρ(K) = 7+√17 16 ≈0.6955. Hence D2= 1 + log(0.6955 ...) log 3 ≈1−0.3634 1.0986 ≈0.669. This nontrivial value is certified by the blur protocol despite first-level overlaps that complicate direct covering arguments. 5.4 What has been achieved • We avoided delicate cylinder coverings or exact transversality. Positivity of the blur reduced everything to a two-state Perron root. • The slope is read once after establishing monotone/positive recursion, aligning with the lawful-switch doctrine. • The method generalizes: any finite-type or graph-directed overlap pattern lifts to a positive matrix M(s); the correlation dimension is certified by solving ρ(M(D2))=rD2−d. 5 6 Outlook: beyond the case study Self-affine carpets with mild shear. Insert a Gaussian in the log-scale transfer operator. The spectral radius as a function of s crosses one at D2 ; positivity enforces uniqueness. Under mild standard hypotheses, correlation dimension agrees with Hausdorff. Bernoulli convolutions at delicate parameters. The L2 heat-blur creates a positive renewal over digit pairs. Even when exact overlaps occur, the protocol yields sharp upper bounds and often the exact D2without diophantine crutches. Weierstrass-type graphs (Minkowski route). The blurred perimeter M ( ε )respects approximate self-affinity; a positive multi-scale inequality pins the Minkowski slope. Borderline parameters (hard for coverings) become tractable. Takeaway. “Dimension is a Mellin slope under positive blur.” Using a single legal switch and a monotone blurred functional, one can certify fractal dimensions in regimes where classical coverings stall. The case study here shows the full program in a small overlapping system; the same template scales to broader families with graphor operator-level positivity. References [1] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd ed., John Wiley & Sons, 2014. [2] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995. [3] Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, 1997. [4] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. [5] R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. [6] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, 2003. [7] P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett. 50 (1983), no. 5, 346–349. [8] Y. Peres and B. Solomyak, Existence of Lq densities for self-similar measures, Indiana Univ. Math. J. 49 (2000), no. 4, 1603–1621. [9] M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180 (2014), no. 2, 773–822. [10] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, 1989. [11] M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2nd ed., Springer Monographs in Mathematics, Springer, 2013. 6 [12] R. S. Strichartz, Self-similarity in harmonic analysis, J. Fourier Anal. Appl. 1(1994), no. 1, 1–37. [13] A. Perišić, Axiom of Blurred Choice, Zenodo, 2025. [14] A. Perišić, Blur as a Universal Principle: Number Theory, Probability, Dynamics, Zenodo, 2025. [15] A. Perišić, Multiplicative Closure of the World, Zenodo, 2025. [16] A. Perišić, Soft-Addition and Soft-Multiplication and the Channel–Switch Error, Zenodo, 2025. 7