All-directions projections of self-affine sets

May 22, 2026

Marstrand-Mattila and the question of exceptions

The Marstrand-Mattila projection theorem describes how Hausdorff dimension behaves under orthogonal projection onto lines. For a Borel set \(X \subset \mathbb{R}^d\), the projection of \(X\) onto almost every line \(V \in\) real projective space \(\mathbb{RP}^{d-1}\) has Hausdorff dimension either equal to that of \(X\), when \(X\) has dimension at most \(1\), or equal to \(1\), the maximal value possible for a subset of a line. The conclusion holds in typical directions and is silent about any particular line.

Theorem 1 (Marstrand (1954) and Mattila (1975)). Let \(X \subset \mathbb{R}^d\) be a Borel set. Then

\[\mathrm{dim}_{\mathrm{H}}(\mathrm{proj}_V(X)) = \min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\]

for \(\gamma_{d,1}\)-almost all \(V \in \mathbb{RP}^{d-1}\), where \(\gamma_{d,1}\) is the Haar measure on \(\mathbb{RP}^{d-1}\).

For sets without internal symmetries this is the whole story. For self-affine sets it is not: the defining iterations encode linear data into certain distinguished directions, and a single line can see this correlation and project the set worse than typical. For a Bedford-McMullen carpet generated by maps with diagonal linear parts, the coordinate axes are themselves exceptional in \(\mathbb{RP}^1\), and the projection in one of those directions can have strictly smaller dimension than the bound in Theorem 1 predicts.

The natural question is when the exceptional set can be eliminated entirely. Reducibility—the existence of a common invariant subspace—generally forces exceptions to persist. The relevant positive condition is the opposite: the linear parts have no shared algebraic structure at all. In dimension two, the appropriate version is that the tuple is proximal and strongly irreducible. In dimension \(d \ge 3\), this is not enough: a contractive proximal strongly irreducible tuple can still preserve an indefinite quadratic form and live inside the corresponding orthogonal group. The condition that survives is Zariski density of the linear parts.

This post is based on a talk I gave at IMPAN, Warsaw, at the conference “Ergodic theory, fractal geometry and Diophantine approximation”.

Corollaries of the main result

Two corollaries of the main set-projection theorem (Theorem 7 below) stand out. The first eliminates the exceptional set under purely algebraic hypotheses on the linear parts, with no separation condition on the translations.

Corollary 2. Let \(X \subset \mathbb{R}^2\) be a self-affine set whose associated tuple \(\mathsf{A}\) is proximal and strongly irreducible. Then

\[\mathrm{dim}_{\mathrm{H}}(\mathrm{proj}_V(X)) = \min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\]

for all \(V \in \mathbb{RP}^1\).

The second is a partial answer to a longstanding folklore question:

Corollary 3. Let \(X \subset \mathbb{R}^d\) be a self-affine set with \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\) whose associated tuple is Zariski dense. Then the Minkowski dimension of \(X\) exists and equals \(\mathrm{dim}_{\mathrm{H}}(X)\).

These extend the planar dimension results of Bárány-Hochman-Rapaport (2019) and Hochman-Rapaport (2022)—removing the strong open set condition of the former and the exponential separation hypothesis of the latter—and carry the all-directions phenomenon to dimension \(d \ge 3\) under the clean algebraic condition of Zariski density. The rest of this post explains the framework.

Background and notation

An affine iterated function system on \(\mathbb{R}^d\) is a tuple \(\Phi = (\varphi_1,\ldots,\varphi_N)\) of contractive affine maps \(\varphi_i(x) = A_ix + t_i\) with \(A_i \in \mathrm{GL}_d(\mathbb{R})\) and \(\|A_i\|<1\). By Hutchinson (1981), there is a unique non-empty compact \(X \subset \mathbb{R}^d\) with

\[X = \bigcup_{i=1}^N \varphi_i(X),\]

the self-affine set of \(\Phi\). We write \(\mathsf{A} = (A_1,\ldots,A_N) \in \mathrm{GL}_d(\mathbb{R})^N\) and \(A_\mathtt{i} = A_{i_1} \cdots A_{i_n}\) for words \(\mathtt{i} = i_1 \cdots i_n\). The orthogonal projection onto a line \(V \in \mathbb{RP}^{d-1}\) is denoted \(\mathrm{proj}_V\).

Two notions of dimension matter. The Hausdorff dimension of \(X\) is denoted \(\mathrm{dim}_{\mathrm{H}}(X)\). The upper Minkowski dimension is

\[\overline{\mathrm{dim}}_{\mathrm{M}}(X) = \limsup_{r \downarrow 0} \frac{\log N_r(X)}{-\log r},\]

with \(N_r(X)\) the maximum number of disjoint balls of radius \(r\) centered in \(X\); the lower Minkowski dimension \(\underline{\mathrm{dim}}_{\mathrm{M}}(X)\) is defined similarly but with \(\liminf_{r \downarrow 0}\). The Minkowski dimension \(\mathrm{dim}_{\mathrm{M}}(X)\) exists when the corresponding \(\liminf_{r \downarrow 0}\) and \(\limsup_{r \downarrow 0}\) coincide. We always have \(\mathrm{dim}_{\mathrm{H}}(X) \le \underline{\mathrm{dim}}_{\mathrm{M}}(X) \le \overline{\mathrm{dim}}_{\mathrm{M}}(X)\), and all three are bounded above by Falconer’s (1988) affinity dimension \(\mathrm{dim}_{\mathrm{aff}}(\mathsf{A})\).

Three algebraic conditions on \(\mathsf{A}\) recur. The tuple is \(k\)-proximal if some product \(A_\mathtt{i}\) acts on \(\wedge^k \mathbb{R}^d\) with a simple dominant eigenvalue. It is strongly \(k\)-irreducible if no finite collection of proper subspaces of \(\wedge^k\mathbb{R}^d\) is left invariant under all \(A_i^{\wedge k}\). The exponential separation condition asks that, infinitely often, distinct length-\(n\) words yield maps separated by at least \(c^n\).

A concrete example to keep in mind throughout the post is a planar IFS \(\Phi_0 = (\varphi_1,\varphi_2)\) with

\[A_1 = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/3 \end{pmatrix}, \qquad A_2 = R_\theta \begin{pmatrix} 1/3 & 0 \\ 0 & 1/2 \end{pmatrix},\]

where \(R_\theta\) is rotation by an irrational multiple of \(\pi\) and \(t_1, t_2 \in \mathbb{R}^2\) are arbitrary translations placing \(X\) in a bounded region. The matrix \(A_1\) has distinct positive eigenvalues and is therefore proximal; the irrational rotation in \(A_2\) rules out any finite collection of invariant lines, so \(\mathsf{A}_0\) is strongly irreducible. Whether \(\Phi_0\) satisfies any separation condition is a question about \(t_1, t_2\), and the central point is that our main results do not care about the answer.

The planar story so far

In the plane the all-directions question has a clear genealogy. Falconer-Kempton (2017) showed that for many planar self-affine sets whose linear parts \(A_i\) have strictly positive entries, the projection has maximal dimension in every direction—with at most one exception, and that exception can occur only if all the \(A_i\) share a common dominant eigenvector, in which case the line along that eigenvector is the unique candidate exceptional direction. When no such common eigenvector exists, the conclusion already holds in every direction.

The breakthrough came with the dimension theorem for planar self-affine sets:

Theorem 4 (Bárány-Hochman-Rapaport (2019)). Let \(X \subset \mathbb{R}^2\) be a strongly irreducible proximal self-affine set satisfying the strong open set condition. Then

\[\mathrm{dim}_{\mathrm{H}}(\mathrm{proj}_V(X)) = \min\{1,\mathrm{dim}_{\mathrm{H}}(X)\} = \min\{1,\mathrm{dim}_{\mathrm{aff}}(\mathsf{A})\}\]

for all \(V \in \mathbb{RP}^1\).

The argument behind Theorem 4 establishes exact-dimensionality of projected self-affine measures, building on Hochman’s (2014) inverse theorem for entropies of convolutions and the local-entropy-averages technique of Hochman-Shmerkin (2012), together with a Furstenberg-measure analysis in the spirit of Hochman-Solomyak (2017). It eliminates the exceptional set, at the cost of the strong open set condition. The subsequent paper of Hochman-Rapaport (2022) relaxed this hypothesis to exponential separation, much closer to a no-overlap condition than to a no-touching condition.

The reducible side of the plane has its own thread. For self-affine measures on carpets, Bárány-K-Pyörälä-Wu (2023+) showed that there are at most two exceptional directions, and Pyörälä (2025) removed the separation hypothesis from the carpet case. The story of an irreducible vs. reducible dichotomy is genuine: by Allen-K-Prokaj-Simon-Troscheit (2026+), reducibility together with a failure of strong irreducibility can already prevent exact-dimensionality of projected measures.

For the example \(\Phi_0\), Theorem 4 and its successors apply only after one assumes the strong open set or exponential separation condition on the translations. Corollary 2 above requires no separation condition at all—and that is the planar contribution of this work.

Zariski density as the right higher-dimensional condition

Strong irreducibility and proximality are projective-side conditions: they constrain the action on \(\mathbb{RP}^{d-1}\), on \(\wedge^k\mathbb{R}^d\), and on subspaces. In dimension two, the projective side is the whole of the algebraic picture: once a contractive tuple is proximal and strongly irreducible, no other invariant structure can remain, and the semigroup it generates is dense in \(\mathrm{GL}_2(\mathbb{R})\) in a strong sense. In higher dimensions a proximal strongly irreducible contractive tuple can still preserve an algebraic structure of a different kind—for example, a quadratic form—and the semigroup can lie inside a proper algebraic subgroup. The condition that catches both is:

Definition 5. A tuple \(\mathsf{A} = (A_1,\ldots,A_N) \in \mathrm{GL}_d(\mathbb{R})^N\) is Zariski dense if the semigroup \(\mathcal{S}(\mathsf{A}) = \langle A_1,\ldots,A_N \rangle\) is dense in \(\mathrm{GL}_d(\mathbb{R})\) in the Zariski topology; equivalently, every polynomial in the matrix entries and \(1/\det\) that vanishes on \(\mathcal{S}(\mathsf{A})\) vanishes on all of \(\mathrm{GL}_d(\mathbb{R})\).

Zariski density is the algebraic analogue of “no missing direction”: no nontrivial polynomial relation holds throughout the semigroup. It implies strong \(k\)-irreducibility and \(k\)-proximality for all \(k \in \{1,\ldots,d-1\}\), and it is an open and dense condition on \(\mathrm{GL}_d(\mathbb{R})^N\)—so for a generic choice of contractive matrices it holds automatically.

In dimension two, Zariski density coincides with what one already had:

Proposition 6. If \(\mathsf{A} = (A_1,\ldots,A_N) \in \mathrm{GL}_2(\mathbb{R})^N\) is contractive, then \(\mathsf{A}\) is Zariski dense if and only if \(\mathsf{A}\) is proximal and strongly irreducible.

The equivalence in Proposition 6 rests on two facts about \(\mathrm{GL}_2(\mathbb{R})\). The proper algebraic subgroups of \(\mathrm{PGL}_2(\mathbb{R})\) are very few—diagonal, upper triangular, \(\mathrm{SO}(2)\), and finite groups—and strong irreducibility plus proximality rules out each. Contractivity then forces the determinants of \(\mathcal{S}(\mathsf{A})\) to be Zariski dense in \(\mathbb{R} \setminus \{0\}\), lifting the projective density to all of \(\mathrm{GL}_2(\mathbb{R})\). In dimension three this lifting can fail: a contractive proximal strongly irreducible tuple can preserve an indefinite quadratic form and remain confined to its orthogonal group up to scaling.

The main set-projection theorem is then:

Theorem 7. Let \(X \subset \mathbb{R}^d\) be a self-affine set whose associated tuple \(\mathsf{A}\) is Zariski dense. Then

\[\mathrm{dim}_{\mathrm{H}}(\mathrm{proj}_V(X)) = \min\{1,\mathrm{dim}_{\mathrm{H}}(X)\} = \min\{1,\overline{\mathrm{dim}}_{\mathrm{M}}(X)\}\]

for all \(V \in \mathbb{RP}^{d-1}\).

Theorem 7 assumes no separation condition on the translations. For the example \(\Phi_0\), Proposition 6 certifies Zariski density, so Theorem 7 applies however much the images \(\varphi_1(X)\) and \(\varphi_2(X)\) overlap, and yields Corollary 2 as a special case.

The proof in three steps

The proof of Theorem 7 does not attack the set directly—it routes through self-affine measures and reduces to subsystems.

The first step is the measure theorem (Theorem 8 below). For a fully supported Bernoulli measure \(\nu\) on the symbolic space \(\mathcal{I}^{\mathbb{N}}\), the self-affine measure \(\mu = \pi_*\nu\) on \(X\) has a Lyapunov dimension \(\mathrm{dim}_{\mathrm{L}}(\nu)\) determined by the Kolmogorov-Sinai entropy of \(\nu\) and the Lyapunov exponents of \(\mathsf{A}\) with respect to \(\nu\):

Theorem 8. Let \(X \subset \mathbb{R}^d\) be a self-affine set satisfying the exponential separation condition, \(\nu\) a fully supported Bernoulli measure, and \(\mu\) the associated self-affine measure. If \(\mathsf{A}\) is \(k\)-proximal and strongly \(k\)-irreducible for all \(k \in \{1,\ldots,d-1\}\), then

\[\mathrm{dim}((\mathrm{proj}_V)_*\mu) = \min\{1,\mathrm{dim}_{\mathrm{L}}(\nu)\}\]

for all \(V \in \mathbb{RP}^{d-1}\).

The proof of Theorem 8 splits into two stages. The first is the Furstenberg-typical projection theorem, which combines two existing inputs. The entropy-dimension formula of Rapaport (2024) gives the projection identity at every \(V \in \mathbb{RP}^{d-1}\), but as a statement about entropy dimension rather than Hausdorff dimension. The exact-dimensionality theorem of Feng (2023) equates the two dimensions for \((\mathrm{proj}_V)_*\mu\) at \(\mu_F\)-almost every \(V\), where \(\mu_F\) is the Furstenberg measure—the unique stationary measure on \(\mathbb{RP}^{d-1}\) associated to \(\mathsf{A}\) and \(\nu\), whose existence and uniqueness follow from \(1\)-proximality and strong \(1\)-irreducibility. Combined, these inputs yield the conclusion of Theorem 8 for \(\mu_F\)-typical \(V\). The new content is the upgrade from \(\mu_F\)-almost every direction to every direction. Rapaport’s entropy-dimension identity already holds at every \(V\), but does not by itself supply the Hausdorff dimension lower bound at a prescribed \(V\). We obtain that bound by verifying the local-entropy-averages hypothesis of Hochman-Shmerkin (2012) at each fixed \(V\) through a suspension semiflow over \(\mathcal{I}^{\mathbb{N}} \times \mathbb{RP}^{d-1}\) whose direction coordinate converges to the Furstenberg direction independently of the starting \(V\). The flow transports Rapaport’s typical-direction estimate to the prescribed \(V\) along symbolic partitions on which \(\mathrm{proj}_V \pi\) has controlled diameter.

The second step is the subsystem construction. To go from measures (which need exponential separation) to sets (which do not), we extract subsystems of \(\Phi\) that satisfy the strong separation condition while preserving the projective data. This proceeds in two stages. The first stage builds a dominated subsystem: starting from a \(1\)-proximal and strongly \(1\)-irreducible tuple, any finite family of words can be sandwiched between prefix and suffix words of bounded length so that the resulting tuple has a common strongly invariant multicone \(\mathcal{C} \subset \mathbb{RP}^{d-1}\). The construction generalizes the planar version of Bárány-Jordan-K-Rams (2021) to arbitrary dimension; its central technical step uses that, under strong \(1\)-irreducibility alone, an invariant subset of \(\mathbb{RP}^{d-1}\) cannot be covered by finitely many proper linear subspaces. The second stage is where Zariski density enters. A packing argument inside the dominated subsystem selects words whose images in \(\mathbb{R}^d\) are pairwise disjoint, keeping the affinity dimension close to \(\overline{\mathrm{dim}}_{\mathrm{M}}(X)\); a finite replacement step then upgrades the selected subsystem to one that is \(k\)-proximal and strongly \(k\)-irreducible for every \(k \in \{1,\ldots,d-1\}\), the hypotheses required by Theorem 8. Two consequences of Zariski density power this upgrade: the existence of a strongly pinching product \(A_\mathtt{i} \in \mathcal{S}(\mathsf{A})\)—a single word whose exterior powers \(A_\mathtt{i}^{\wedge k}\) are simultaneously pinching for all \(k\)—and the fact that the semigroup contains words whose exterior powers avoid any prescribed finite list of proper subspaces.

The third step is the transfer. Apply Theorem 8 to the strongly separated subsystem with a Bernoulli measure of Lyapunov dimension close to \(\min\{1,\overline{\mathrm{dim}}_{\mathrm{M}}(X)\}\). Monotonicity of Hausdorff dimension under inclusion of the subsystem’s attractor in \(X\) promotes the conclusion back to the original set.

Minkowski dimension below one

Self-affine sets are the simplest setting where the Minkowski dimension is not known to exist. For self-similar sets it does, by Falconer (1989), and the same is true for sub-self-conformal sets by Feng-Hu (2009), with no separation conditions. For self-affine carpets, \(\mathrm{dim}_{\mathrm{H}}\) and \(\overline{\mathrm{dim}}_{\mathrm{M}}\) generically differ—and the upper Minkowski dimension often equals the affinity dimension while the Hausdorff dimension does not—but in all known examples the upper and lower Minkowski dimensions still coincide. No finitely generated self-affine set is known for which \(\underline{\mathrm{dim}}_{\mathrm{M}} < \overline{\mathrm{dim}}_{\mathrm{M}}\), and the question is open in every dimension. The cleanest source of partial answers is via projection, and Theorem 7 is the input we need.

The proof of Corollary 3 is four lines. The general Lipschitz bound \(\mathrm{dim}_{\mathrm{H}}(\mathrm{proj}_V(X)) \le \min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\), combined with Theorem 7 and the hypothesis \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\), yields

\[\underline{\mathrm{dim}}_{\mathrm{M}}(X) \ge \mathrm{dim}_{\mathrm{H}}(X) \ge \mathrm{dim}_{\mathrm{H}}(\mathrm{proj}_V(X)) = \min\{1,\overline{\mathrm{dim}}_{\mathrm{M}}(X)\} = \overline{\mathrm{dim}}_{\mathrm{M}}(X),\]

so the two Minkowski dimensions coincide with \(\mathrm{dim}_{\mathrm{H}}(X)\). The hypothesis \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\) is exactly what restricts the conclusion: above dimension one, the projection bound saturates at \(1\).

Where this leaves us

Several questions stay open. Does the Minkowski dimension exist for every self-affine set? The Zariski-dense case with \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\) is a partial answer, but the higher-dimensional case and the reducible case remain. For Lebesgue-typical translations, Feng-Xie (2025+) and Morris-Sert (2025+) show that the all-directions phenomenon genuinely fails for reducible tuples: there exist reducible self-affine sets for which the upper Minkowski dimension of the projection drops below \(\min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\) in some direction. So Zariski density is doing real work in Theorem 7—but whether it is the minimal algebraic condition, or whether a finer dichotomy organizes the reducible side as well, is the next question on this list.

The other open frontier is whether Theorem 7 extends from projections onto lines to projections onto \(k\)-dimensional subspaces with \(k \ge 2\). The Marstrand-Mattila statement does extend in this form, with \(\min\{k,\mathrm{dim}_{\mathrm{H}}(X)\}\) in place of \(\min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\) for a typical \(k\)-plane; and a \(k=2\) all-directions analogue of Theorem 7 would lift Corollary 3 to the range \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 2\). Our proof, however, does not immediately follow. One obstacle sits inside the packing argument in the second stage of step two: from each packing ball of \(X\) at scale \(r\), the construction selects a single representative cylinder \(\varphi_\mathtt{i}(X)\), contained in an ellipsoid whose longest semiaxis \(\alpha_1(A_\mathtt{i})\) is essentially \(r\). Under \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\), the singular value function controlling the affinity dimension is \(\varphi^s(A) = \alpha_1(A)^s\) for \(s \le 1\), which depends only on the longest semiaxis, and matching the packing radius to that one semiaxis is enough. For projection onto a \(2\)-plane the supplemental range is \(1 < s \le 2\), where \(\varphi^s(A) = \alpha_1(A)\,\alpha_2(A)^{s-1}\) also involves the second-longest semiaxis \(\alpha_2(A)\). The packing of \(X\) records only one scale per ball, and it is genuinely unclear how to choose a single representative cylinder per packing ball that simultaneously controls \(\alpha_2\).