The linear case of the Palis–Takens conjecture and the existence of positive-measure self-similar sets with empty interior on the line

Two linked deterministic questions about self-similar sets on the real line remain open, although their random analogues were settled by Dekking, Simon, Székely and Szekeres (Advances in Mathematics, 2024). First, the interior question posed by Peres and Solomyak (2000): does there exist a self-similar subset of the line — the attractor of a finite iterated function system of contracting similarities — with positive Lebesgue measure and empty interior? By a theorem of Schief, under the open set condition positive measure forces an interval, so the question lives entirely in the overlapping regime; in the plane it was answered affirmatively by Csörnyei, Jordan, Pollicott, Preiss and Solomyak (2006), but on the line no example is known and no nonexistence theorem exists — even for typical translation parameters. Second, the linear case of the Palis–Takens conjecture: is it true, typically, that the arithmetic difference of two linear (self-similar) Cantor sets on the line either contains a non-degenerate interval or has Lebesgue measure zero — in particular, does the difference typically contain an interval once the Hausdorff dimensions sum to more than one? Moreira and Yoccoz (2001) proved the conjecture for generic non-linear dynamically defined Cantor sets, but for linear Cantor sets only restricted perturbative partial results are known. This record states the interior question as its target, in its natural generality, together with the linear Palis–Takens conjecture as a closely related open question and the precise methodological link between them established by the random-case proof.

Background

When a smooth surface map has a homoclinic tangency — a non-transverse intersection between the stable and the unstable manifold of a periodic point — the nearby dynamics is organized by Cantor sets (non-empty compact subsets of the real line that are perfect and totally disconnected, such as the classical middle-thirds Cantor set). Palis and Takens, studying the unfolding of such bifurcations (their 1987 Annals paper and 1993 monograph), were led to a conjecture on the size of sums of Cantor sets: for two Cantor sets $C_1, C_2\subset\mathbb{R}$ defined by dynamics, the algebraic difference $C_2-C_1=\{y-x: x\in C_1, y\in C_2\}$ should typically be either large — contain a non-degenerate interval — or small — have Lebesgue measure zero; in particular, if the Hausdorff dimensions (the standard fractal dimension, denoted $\dim_H$) satisfy $\dim_H C_1+\dim_H C_2>1$, the difference should typically contain an interval. Here and throughout, typically refers to genericity in the relative placement of the two sets; the precise formulations in the literature are Baire-generic or full Lebesgue measure in the relative translation (and, where applicable, scaling) parameters.

The simplest dynamically defined Cantor sets are the self-similar ones. An iterated function system (IFS) on the line is a finite list of contracting similarities $S_i(x)=r_i x+t_i$ with $0<|r_i|<1$; a classical theorem of Hutchinson gives a unique non-empty compact attractor $C_S$ with $C_S=\bigcup_i S_i(C_S)$, called self-similar. The similarity dimension is the unique $s\geq 0$ with $\sum_i |r_i|^s=1$. Under the open set condition (OSC) — the existence of a non-empty open set $O$ with the images $S_i(O)$ pairwise disjoint and contained in $O$ — the Hausdorff dimension of the attractor equals the similarity dimension. Schief (1994) sharpened this on the line: if the OSC holds and $\sum_i |r_i|=1$, then the attractor has positive Lebesgue measure and non-empty interior. Hence any example with positive measure but empty interior must come from overlapping systems, where no separation condition holds and the standard dimension theory (Hutchinson; Hochman's entropy inverse theorems) controls dimension but not the structure of the measure-theoretic interior.

Moreira and Yoccoz (2001) proved the Palis–Takens conjecture for generic pairs of regular Cantor sets defined by genuinely non-linear dynamics, via their stable intersections theorem: generic pairs with $\dim_H C_1+\dim_H C_2>1$ have intersections that persist under all sufficiently small perturbations. The renormalization and limit-geometry machinery of that proof uses nonlinearity of the defining dynamics in an essential way, and the conjecture for generic linear Cantor sets — attractors of finite systems of affine maps, i.e. self-similar sets — has remained open. Dekking, Simon, Székely and Szekeres (2024), who proved the conjecture for randomly perturbed self-similar systems, state explicitly that it has not been proven for generic linear Cantor sets.

The same 2024 paper isolates a second, more elementary deterministic question that forms the engine of its proof and is itself a recognized open problem, posed by Peres and Solomyak in 2000: does there exist a self-similar set on the line with positive Lebesgue measure and empty interior? In the plane the answer is affirmative: Csörnyei, Jordan, Pollicott, Preiss and Solomyak (2006) exhibited an explicit ten-map family of similarities of ratio $1/3$ of the plane for which almost every choice of two translation parameters yields positive measure with empty interior, and Baker and Sidorov (2019) gave an explicit example for an infinitely generated system. On the line, however, no example is known and no theorem rules one out, even for Lebesgue-typical translation parameters of a fixed vector of contraction ratios: Saglietti, Shmerkin and Solomyak (2018) proved that for almost every choice of translations, similarity dimension above one gives positive Lebesgue measure, but the interior question stayed beyond reach. Keane, Smorodinsky and Solomyak (1995) proved non-empty interior above an explicit dimension threshold $s_0>1$ for the family of sets of numbers admitting base-$\gamma$ expansions using only the digits $\{0,1,3\}$, and Dong and Shang (2010) showed that positive measure forces an interval for two restricted classes (systems of two similarities and uniformly discrete digit systems); the general overlapping case is untouched.

For randomly perturbed self-similar systems — the model of Jordan, Pollicott and Simon (2007), where small independent random translations are added at every construction step — the analogue of the interior question was settled by Dekking et al. (2024): if the similarity dimension exceeds one, the random attractor has non-empty interior almost surely, and as a corollary the Palis–Takens dichotomy holds for two independent copies whenever the similarity dimension exceeds one half. Later work strengthened the random theory (Bárány–Rams 2025: Hölder-continuous densities of random self-similar measures; Baker–Koivusalo–Troscheit–Zhang 2026: random Bernoulli convolutions, i.e. laws of infinite random signed power series with randomly drawn contraction ratios, also have interior almost surely), and Orgoványi and Simon (2026) showed that a different randomization (overlapping Mandelbrot percolation on the line) produces, almost surely, sets with positive measure and empty interior — while explicitly noting that the deterministic line case is currently unknown. The two deterministic questions stated below are exactly the ones these random theorems mirror, and they are linked methodologically: in the random proof the single-set interior statement is the main theorem from which the two-set dichotomy is derived.

Problem Statement

Let $K\subset\mathbb{R}$ be the attractor of a finite iterated function system of contracting similarities of the real line (a self-similar set); no separation condition such as the open set condition is assumed. The target problem is the following deterministic interior question, which is open; the closely related second deterministic question, the linear Palis–Takens conjecture, is stated after it as context.

(Q1 — the Peres–Solomyak interior question on the line.) Does there exist such a self-similar set $K$ with positive Lebesgue measure and empty interior? Equivalently, either some finite system of contracting similarities has an attractor with positive Lebesgue measure and empty interior, or every self-similar subset of $\mathbb{R}$ with positive Lebesgue measure contains a non-degenerate interval. The question is open even in its typical-parameter form: for a fixed vector of contraction ratios with $\sum_i |r_i|>1$, it is unknown whether for Lebesgue-almost every (or even Baire-generic) choice of the translation parameters the attractor, when it has positive measure, must contain an interval.

(Q2 — the linear case of the Palis–Takens conjecture; a related open question, not part of the target.) Let $C_1, C_2\subset\mathbb{R}$ be Cantor sets that are attractors of finite systems of contracting affine maps (linear Cantor sets, i.e. possibly non-homogeneous self-similar Cantor sets). Is it true, in any of the standard precise senses of typicality for the relative placement (Baire-generic or Lebesgue-almost every relative translation, and where applicable scaling), that the arithmetic difference $C_2-C_1$ either contains a non-degenerate interval or has Lebesgue measure zero; in particular, if $\dim_H C_1+\dim_H C_2>1$, does $C_2-C_1$ typically contain an interval? The conjecture concerns the natural class of all linear Cantor sets on the line, without separation, uniformity or digit restrictions.

The link between the two questions, as in the source: Q1 is the single-set interior statement that drives the proof in the random case — Dekking et al. (2024) prove the random analogue of Q1 (non-empty interior almost surely when the similarity dimension exceeds one) as their main theorem and derive the random Palis–Takens statement from it. Resolving Q1 deterministically is the recognized central step toward Q2. The target problem is Q1, asked in full generality (no separation condition assumed); Q2 is stated only as closely related context.

References

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