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arxiv: 2604.19986 · v1 · submitted 2026-04-21 · 🧮 math.DS · math.MG

On the intersections of homogeneous self-similar sets with their translates in mathbb{R}^(n) and a formulation of multiplicative invariance in mathbb{Z}^(n)

Neil MacVicar This is my paper

Pith reviewed 2026-05-10 00:43 UTC · model grok-4.3

classification 🧮 math.DS math.MG
keywords self-affine setsself-similar setsfractal intersectionsiterated function systemsmultiplicative invariancetorus dynamicsCantor setHausdorff dimension
0
0 comments X

The pith

Translations of homogeneous self-similar sets in R^n produce self-affine intersections under algebraic conditions on the vector, with explicit dimension improvements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes the classical intersections of the middle-third Cantor set with its translates to homogeneous self-affine attractors in higher-dimensional Euclidean space. It supplies necessary and sufficient conditions on the translation vector that guarantee the intersection remains self-affine for a defined class of such attractors. In the self-similar subcase, the mapping from translation to the fractal dimension of the intersection receives sharper treatment than prior one-dimensional results. The work closes with a case study on complex bases and introduces a notion of multiplicative invariance for subsets of Z^n that links directly to invariant sets on the n-torus.

Core claim

For homogeneous self-affine sets generated by iterated function systems, the intersection with a translate is itself self-affine precisely when the translation satisfies algebraic commensurability conditions derived from the linear parts of the maps. When the attractor is self-similar, the Hausdorff dimension of the intersection becomes a more explicitly describable function of the translation than in earlier work, and this framework extends to a definition of multiplicative invariance in Z^n whose invariant sets correspond to those of the n-dimensional torus under the natural action.

What carries the argument

Algebraic conditions on the translation vector α that ensure the intersection is coded by a subsystem of the original iterated function system, preserving self-affinity.

If this is right

  • The dimension of the intersection can be read off from the translation without exhaustive enumeration of overlaps.
  • The complex-base case study yields concrete dimension values for specific translations in the plane.
  • Multiplicative invariance supplies a lattice-level characterization that matches known one-dimensional correspondences with torus invariants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The algebraic conditions could serve as a template for checking self-affinity in numerically generated fractals arising from linear contractions.
  • The torus linkage may allow transfer of ergodic properties from the lattice setting back to the fractal intersections.
  • Extensions to non-homogeneous attractors might be tested by perturbing the linear parts while preserving the intersection coding.

Load-bearing premise

The self-affine sets must belong to the restricted homogeneous class where intersections can be tracked uniformly through the symbolic dynamics of the iterated function system.

What would settle it

An explicit homogeneous self-affine set together with a translation vector that satisfies the stated algebraic conditions yet produces an intersection that is not self-affine, or a self-similar example where the computed dimension deviates from the improved formula.

Figures

Figures reproduced from arXiv: 2604.19986 by Neil MacVicar.

Figure 1
Figure 1. Figure 1: The Sierpinski Triangle . . . . . . . . . . . . . . . . . . . 2 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: The Sierpinski Triangle If geometry is the study of shapes, then fractal geometry might be called the study of shapes that exhibit complexity at all scalings. Consider your favourite polygon. If we zoom in on a point on the boundary of the polygon, we eventually lose sight of the polygon. A sufficiently small neighbourhood of a point on an edge will only capture a line segment. If the point is a vertex, … view at source ↗
Figure 1
Figure 1. Figure 1: is the Sierpinski triangle. Its construction begins with an equi [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The set T2 to bound the set Tn := Tn,{0,1,...,n2} in a ball and then discern the horizontal translations that ensure the balls do not intersect. Consider the approximations of Tn when n = 2, 3, 4 contained in Figures 2.1, 2.2, and 2.3. In each of Figures 2.1, 2.2, and 2.3, the complement of Tn within a ball containing it is large. In Chapter 6, we lower the separation condition by refining our understand… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The set T3 [PITH_FULL_IMAGE:figures/full_fig_p025_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The set T4 19 [PITH_FULL_IMAGE:figures/full_fig_p025_2_3.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: The neighbour graph for (10, {0, 1, . . . , 9})-representations. 9 have a difference of ±9. The path that never leaves the 0 vertex corresponds to an indistinct pair of representations. The neighbour graph that governs the equivalence of (−n+i, {0, 1, . . . , n2})- representations is given in [16]. It carries additional information by keeping track of three states at each vertex that are associated with … view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: The neighbour graph for (−n + i, {0, 1, . . . , n2})-representations, n ≥ 3. 108 [PITH_FULL_IMAGE:figures/full_fig_p114_6_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: starting at state [PITH_FULL_IMAGE:figures/full_fig_p115_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: is the path that moves along the states [PITH_FULL_IMAGE:figures/full_fig_p116_6.png] view at source ↗
read the original abstract

This thesis generalizes the study of $C\cap(C + \alpha)$ where $C$ is the middle third Cantor set to self-affine sets in $\mathbb{R}^{n}$. We present sufficient and necessary conditions for when the translation $\alpha$ produces a self-affine intersection for a particular class of self-affine sets. In the case where the attractor is self-similar, we improve results concerning the function from $\alpha$ to the fractal dimension of the intersection. This lends itself to a case study of the complex number system $(-n + i, \{0, 1, . . . , n^{2}\})$, when $n$ is an integer greater than or equal to $2$. Lastly, we present a definition of multiplicative invariance for subsets of $\mathbb{Z}^{n}$ and establish a connection, known in the one-dimensional case, between them and invariant sets of the $n$-dimensional torus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript generalizes the study of intersections C ∩ (C + α) for the middle-third Cantor set to homogeneous self-affine sets in R^n. It claims to establish sufficient and necessary conditions under which a translation α yields a self-affine intersection for a restricted class of homogeneous self-affine attractors. In the self-similar case it asserts improvements to the function mapping α to the fractal dimension of the intersection. A case study is given for the complex base (-n + i) with digits {0, …, n²} when n ≥ 2. The paper also introduces a definition of multiplicative invariance for subsets of Z^n and links it to invariant sets on the n-torus, extending the known one-dimensional correspondence.

Significance. If the stated conditions are rigorously derived from the overlap analysis and the dimension-function improvement is shown to be strictly stronger than existing results without circular reduction to prior definitions, the work would usefully extend fractal-intersection techniques to higher dimensions and supply a natural n-dimensional analogue of multiplicative invariance. The explicit scoping to a homogeneous class and the concrete complex-base example are positive features that keep the claims falsifiable within the declared setting.

major comments (2)
  1. [Abstract, §1] Abstract and §1: the claim of 'sufficient and necessary conditions' for self-affine intersections is load-bearing for the central contribution, yet the abstract supplies neither the explicit statement of the conditions nor the derivation; the reader cannot verify whether the conditions reduce to the 1D overlap criterion or introduce new restrictions on the contraction ratios or separation properties.
  2. [§3 (self-similar case)] The asserted improvement to the dimension function α ↦ dim(C ∩ (C + α)) for self-similar attractors must be shown to be non-circular; if the improvement is obtained solely by re-using the same overlap data already employed in the 1D case, the gain in generality should be quantified against the prior literature cited in §2.
minor comments (3)
  1. [Title, Abstract] The title refers exclusively to 'homogeneous self-similar sets' while the abstract and introduction repeatedly invoke 'self-affine sets'; a brief clarifying sentence distinguishing the two classes and stating which results apply to each would improve readability.
  2. [Case study section] Notation for the attractor, the translation α, and the digit set in the complex-base case study should be introduced once and used consistently; several passages in the case-study section appear to reuse symbols without re-definition.
  3. [Final section] The definition of multiplicative invariance in Z^n is presented as a direct analogue of the 1D torus correspondence; a short remark comparing the new definition with any existing n-dimensional notions in the literature would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments that will help improve its clarity. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: the claim of 'sufficient and necessary conditions' for self-affine intersections is load-bearing for the central contribution, yet the abstract supplies neither the explicit statement of the conditions nor the derivation; the reader cannot verify whether the conditions reduce to the 1D overlap criterion or introduce new restrictions on the contraction ratios or separation properties.

    Authors: We agree with this observation. The abstract will be revised to explicitly state the sufficient and necessary conditions for self-affine intersections, which are based on the overlap analysis for the homogeneous class. These conditions extend the 1D criterion by incorporating restrictions on the contraction ratios of the affine maps and the separation properties in R^n. The derivation is detailed in Section 2, and we will include a pointer to it in the abstract for easy verification. revision: yes

  2. Referee: [§3 (self-similar case)] The asserted improvement to the dimension function α ↦ dim(C ∩ (C + α)) for self-similar attractors must be shown to be non-circular; if the improvement is obtained solely by re-using the same overlap data already employed in the 1D case, the gain in generality should be quantified against the prior literature cited in §2.

    Authors: The improvement is not circular, as it generalizes the dimension calculation to the n-dimensional self-similar setting, allowing for attractors defined by multiple simultaneous contractions that cannot be reduced to independent 1D cases. We build on but do not solely reuse the 1D overlap data; the new dimension function in §3 captures interactions across dimensions. To address the quantification, we will add a subsection comparing our results to the literature in §2, highlighting the additional generality for examples like the complex base case study. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper generalizes 1D Cantor set intersection results to a restricted class of homogeneous self-affine and self-similar attractors in R^n by deriving sufficient and necessary conditions on translations alpha that produce self-affine intersections. Dimension-function improvements for the self-similar case are obtained directly from overlap analysis within the declared class, without fitted parameters or reductions to prior definitions. The multiplicative invariance definition on Z^n is introduced as a direct n-dimensional analogue of the known 1D torus correspondence, forming an independent extension rather than a renaming or self-referential step. No load-bearing equations, self-citations, or ansatzes reduce the claimed results to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The work appears to rest on standard assumptions of self-similar/self-affine set theory.

pith-pipeline@v0.9.0 · 5470 in / 1194 out tokens · 21794 ms · 2026-05-10T00:43:57.523755+00:00 · methodology

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Reference graph

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