All-directions projections of self-affine sets

July 17, 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 the 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 a general Borel set this is the whole story. The almost-every quantifier cannot be improved: any prescribed line \(V\) is exceptional for some set, since a set of positive dimension contained in the orthogonal complement \(V^\perp\) projects onto \(V\) to a single point. Without further structure on \(X\), no individual direction can be ruled out as exceptional.

Self-affine sets carry internal symmetries that change the situation. Such a set is a finite union of affine copies of itself and hence, upon iteration, a union of affine copies of itself at all scales; and these symmetries are visible to projections, because projecting an affine copy of \(X\) onto a line \(V\) agrees, up to an invertible affine map between the two lines, with projecting \(X\) itself onto the line \(A^\top V\), where \(A\) is the linear part of the defining map. The projection in one direction is therefore assembled from affine copies of the projections in the moved directions, so information can propagate from typical directions to any fixed one, and one can hope that for self-affine sets the Marstrand-Mattila theorem holds with no exceptional directions at all.

Some assumption is nevertheless necessary, since the internal symmetries can also work against the projections: for a Bedford-McMullen carpet, generated by maps with diagonal linear parts, the coordinate axes are invariant under every move \(V \mapsto A^\top V\), and the projection onto a coordinate axis is a self-similar set whose dimension can be strictly smaller than Theorem 1 predicts, so the axes can be genuinely exceptional directions.

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 conditions live on the projective side: they ask the linear parts to mix directions so thoroughly that no finite family of subspaces is preserved. In dimension two, strong irreducibility of the tuple alone suffices, and this is sharp. In dimension \(d \ge 3\), the hypotheses that power the result are strong pinching together with strong irreducibility in every exterior power; they hold for a typical tuple of matrices, since both follow from Zariski density of the linear parts.

This post is based on a talk I gave on May 22, 2026, at IMPAN, Warsaw, during the conference “Ergodic theory, fractal geometry and Diophantine approximation”. It explains our recent paper with Balázs Bárány and István Kolossváry.

Two consequences of the main result

Two corollaries of the main set-projection theorem (Theorem 7 below) stand out. The first eliminates the exceptional set in the plane under strong irreducibility alone—with no separation condition on the translations and no proximality assumption—and this hypothesis is sharp.

Corollary 2 (Bárány-K-Kolossváry (2026+)). Let \(X \subset \mathbb{R}^2\) be a self-affine set whose associated tuple \(\mathsf{A}\) is strongly irreducible. 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 \(V \in \mathbb{RP}^1\).

The second is a partial answer to a long-standing folklore question:

Corollary 3 (Bárány-K-Kolossváry (2026+)). Let \(X \subset \mathbb{R}^d\) be a self-affine set with \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\) whose associated tuple is strongly pinching and strongly \(k\)-irreducible for all \(k \in \{1,\ldots,d-1\}\). 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, with Corollary 2 in addition removing the proximality assumption of both—and carry the all-directions phenomenon to dimension \(d \ge 3\) under hypotheses that hold for a typical tuple of matrices. To our knowledge, Corollary 3 is the first general existence result for the Minkowski dimension of self-affine sets that requires neither a separation condition nor any restriction on the translation vectors. 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 \(\mathrm{dim}_{\mathrm{aff}}(\mathsf{A})\).

Four algebraic conditions on \(\mathsf{A}\) recur. The tuple is \(k\)-proximal if some sequence of normalized products \(c_n A_{\mathtt{i}_n}^{\wedge k}\) converges to a rank-one linear map on \(\wedge^k \mathbb{R}^d\); under \(k\)-irreducibility, this holds exactly when some product \(A_\mathtt{i}^{\wedge k}\) has a simple dominant eigenvalue. It is strongly \(k\)-irreducible if no finite collection of proper non-trivial subspaces of \(\wedge^k\mathbb{R}^d\) is left invariant under all \(A_i^{\wedge k}\). A matrix is pinching if it is diagonalizable and its eigenvalues have pairwise distinct absolute values, and the tuple is strongly pinching if a single product \(A_\mathtt{i}\) has all its exterior powers \(A_\mathtt{i}^{\wedge k}\), \(k \in \{1,\ldots,d-1\}\), pinching; strong pinching implies \(k\)-proximality for every \(k\), and for irreducible planar tuples it is equivalent to proximality. The exponential separation condition asks for a constant \(c > 0\) such that, for every \(n\), distinct length-\(n\) words yield maps separated by at least \(c^n\).

Concrete planar tuples satisfying these algebraic hypotheses are easy to construct. Take a diagonal contraction with distinct positive eigenvalues and pair it with a contraction obtained by composing a diagonal matrix with a rotation through an irrational multiple of \(\pi\). The first matrix is pinching, and every finite family of lines invariant under it consists of its two eigendirections. The rotated matrix moves those directions, so the resulting tuple is strongly pinching and strongly irreducible. The translation vectors may then be chosen arbitrarily, since our main set-projection results impose no separation condition.

The planar story so far

In the plane the all-directions question has a clear genealogy. Falconer-Kempton (2017) showed that for many dominated planar self-affine sets, the projection achieves the dimension \(\min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\) in all but at most one 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 rests on the Ledrappier-Young formula and exact-dimensionality of projected self-affine measures, established by Bárány-K (2017)—a critical component of the proof—together with the local-entropy-averages technique of Hochman-Shmerkin (2012). It eliminates the exceptional set, at the cost of the strong open set condition. As written, however, the projection part of their argument contains a slight gap and yields the conclusion only for dominated and irreducible linear parts; our measure-level analysis explains and closes this gap. The subsequent paper of Hochman-Rapaport (2022) relaxed the separation 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 planar carpets satisfying the strong separation condition and an irrationality assumption on the entries of the linear parts, Bárány-K-Pyörälä-Wu (2023+) showed that there are at most two exceptional directions, and Pyörälä (2025) removed the separation hypothesis. The boundary is delicate: by Allen-K-Prokaj-Simon-Troscheit (2026+), the projection of an irreducible planar ergodic measure that is not strongly irreducible need not be exact-dimensional.

Theorem 4 assumes the strong open set condition, while the subsequent theorem of Hochman-Rapaport requires exponential separation. Corollary 2 requires no separation condition and no proximality assumption: strong irreducibility alone suffices. Its proof splits according to proximality. Proximal tuples fall under the main higher-dimensional theorem, Theorem 7 below, while a non-proximal strongly irreducible planar tuple is conjugate to a system of contracting similarities whose orthogonal parts generate an infinite group, so the projection theorem of Farkas (2016) for self-similar sets finishes the argument. In particular, Corollary 2 recovers the planar case of Farkas’ theorem and complements it with the proximal, genuinely non-conformal, self-affine sets.

Strong irreducibility is also exactly the right hypothesis, as witnessed by a concrete four-map golden-ratio example, Example 6.3. Its linear parts are diagonal and antidiagonal, the system satisfies the strong separation condition, and its tuple is irreducible but not strongly irreducible. Both coordinate projections have Hausdorff dimension \(\log_2(\frac{1+\sqrt{5}}{2}) \approx 0.694\), the base-two logarithm of the golden ratio, while \(\mathrm{dim}_{\mathrm{H}}(X) \ge \tfrac45\). The antidiagonal maps interchange the coordinate axes, so the two coordinate projections coincide as a self-similar set with an exact overlap, and both axes are exceptional directions. Hence strong irreducibility cannot be relaxed to irreducibility in Corollary 2, even under the strong separation condition.

Strong pinching and the role of Zariski density

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 the main set-projection theorem, Theorem 7 below, asks for more of the same projective kind: the tuple should be strongly pinching and strongly \(k\)-irreducible for every \(k \in \{1,\ldots,d-1\}\). The natural worry is that this longer list of hypotheses might be restrictive or hard to check. The reassurance comes from a single algebraic condition that implies them all at once:

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. Every Zariski dense tuple is strongly pinching and strongly \(k\)-irreducible for all \(k \in \{1,\ldots,d-1\}\), and the Zariski dense tuples form a non-empty Zariski open subset of \(\mathrm{GL}_d(\mathbb{R})^N\)—in particular, an open and dense set in the Euclidean topology. For a generic choice of contractive matrices, the hypotheses of Theorem 7 therefore hold automatically. In dimension two, Zariski density coincides with what one already had:

Proposition 6 (Bárány-K-Kolossváry (2026+)). 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 a dichotomy for the algebraic subgroups of \(\mathrm{PGL}_2(\mathbb{R})\): a subgroup that preserves no non-empty finite subset of \(\mathbb{RP}^1\) is either all of \(\mathrm{PGL}_2(\mathbb{R})\) or conjugate into the projective image of the conformal group \(\mathrm{GO}(2)\). Strong irreducibility rules out finite invariant sets, proximality rules out the conformal alternative, and contractivity forces the determinants of \(\mathcal{S}(\mathsf{A})\) to be Zariski dense in \(\mathbb{R} \setminus \{0\}\), lifting the projective density to all of \(\mathrm{GL}_2(\mathbb{R})\). For contractive planar tuples, Zariski density therefore coincides exactly with the hypotheses of Theorem 7 below. In dimension three the coincidence fails: there is a contractive tuple, strongly pinching and strongly \(k\)-irreducible for all \(k\), whose semigroup preserves an indefinite quadratic form up to scalar and is therefore confined to a proper algebraic subgroup. Zariski density is thus strictly stronger than the hypotheses of Theorem 7—which, notably, still covers such quadratic-form-preserving tuples. The main set-projection theorem is then:

Theorem 7 (Bárány-K-Kolossváry (2026+)). Let \(X \subset \mathbb{R}^d\) be a self-affine set whose associated tuple \(\mathsf{A}\) is strongly pinching and strongly \(k\)-irreducible for all \(k \in \{1,\ldots,d-1\}\). 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, and its hypotheses are typical in the strongest sense:

Theorem 8 (Bárány-K-Kolossváry (2026+)). The set of tuples in \(\mathrm{GL}_d(\mathbb{R})^N\) that are strongly pinching and strongly \(k\)-irreducible for all \(k \in \{1,\ldots,d-1\}\) contains an open and dense subset of \(\mathrm{GL}_d(\mathbb{R})^N\).

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 9 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 9 (Bárány-K-Kolossváry (2026+)). 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 9 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 measure on \(\mathbb{RP}^{d-1}\) stationary under the transposes \(A_i^\top\) weighted by \(\nu\), whose existence and uniqueness follow from \(1\)-proximality and strong \(1\)-irreducibility. Combined, these inputs yield the conclusion of Theorem 9 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 \(V\). A stopping-time comparison along the flow then transfers the resulting entropy bounds from the evolving direction to the prescribed \(V\), using 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 a packing argument on \(X\) itself: from a maximal packing of \(X\) at a small scale it selects words whose images in \(\mathbb{R}^d\) are pairwise disjoint and feeds them into the dominated construction, keeping the affinity dimension close to \(\min\{1,\overline{\mathrm{dim}}_{\mathrm{M}}(X)\}\); 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 9. The two standing hypotheses of Theorem 7 power this upgrade directly. Strong pinching supplies a product \(A_\mathtt{i} \in \mathcal{S}(\mathsf{A})\) whose exterior powers \(A_\mathtt{i}^{\wedge k}\) are simultaneously pinching for all \(k\). Strong \(k\)-irreducibility for all \(k\) supplies the simultaneous escape property: a single word of the semigroup can steer any prescribed finite family of subspaces, simultaneously across all the exterior powers, away from any prescribed finite list of proper subspaces. The escape property is in fact equivalent to strong \(k\)-irreducibility for all \(k\).

The third step is the transfer. Apply Theorem 9 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 and, more generally, sub-self-conformal sets it does, by Falconer (1989), with no separation conditions, and Feng-Hu (2009) extended this to attractors of weakly conformal iterated function systems. For self-affine carpets, \(\mathrm{dim}_{\mathrm{H}}\) and \(\overline{\mathrm{dim}}_{\mathrm{M}}\) generically differ, 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? Corollary 3 answers the strongly pinching, strongly irreducible case with \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) \le 1\), but the case \(\overline{\mathrm{dim}}_{\mathrm{M}}(X) > 1\) and the reducible tuples remain.

The reducible case is also where the all-directions phenomenon itself breaks down, and two recent preprints make the boundary precise. Under an irreducibility condition on the linear parts, Feng-Xie (2025+) and Morris-Sert (2025+) extend the Marstrand-Mattila projection theorem for self-affine sets to every subspace, of any dimension, for Lebesgue-almost every choice of translations; the algebraic hypotheses are milder than those of Theorem 7, at the price of typical rather than arbitrary translations. In the plane, Feng-Xie moreover pinpoint where the phenomenon fails for typical translations, and the mechanism is the carpet one: a direction can be deficient only if it is invariant under all the transposed linear parts; the projection of the system in such a direction is self-similar, and the deficiency occurs exactly when the similarity dimension of this projection falls strictly below \(\min\{1,\mathrm{dim}_{\mathrm{aff}}(\mathsf{A})\}\). Feeding such a tuple into the projection bound of Morris-Sert, valid for every choice of translations, and into the classical almost-sure dimension formula of Falconer (1988) then shows that, for Lebesgue-almost every choice of translations, the resulting reducible planar self-affine set has a projection in the invariant direction whose upper Minkowski dimension stays strictly below \(\min\{1,\mathrm{dim}_{\mathrm{H}}(X)\}\).

Both the invariant direction and the associated dimension gap are dictated by the linear parts alone; no choice of translations removes either, so the conclusion of Theorem 7 can genuinely fail without suitable irreducibility assumptions. In the plane the sharpness statement is stronger: strong irreducibility in Corollary 2 cannot be replaced by irreducibility. Indeed, the four-map golden-ratio example above is irreducible but not strongly irreducible, satisfies the strong separation condition, and has both coordinate axes as exceptional directions. In higher dimensions, whether strong pinching together with strong irreducibility in every exterior power is the minimal algebraic condition, or whether a finer dichotomy organizes the reducible side as well, is the next question on this list.

For the existence of the Minkowski dimension, these examples close a route rather than the question. The squeeze in the proof of Corollary 3 runs through the Hausdorff dimension and can only certify existence together with the equality \(\mathrm{dim}_{\mathrm{M}}(X) = \mathrm{dim}_{\mathrm{H}}(X)\); once the Hausdorff dimension lies strictly below the lower Minkowski dimension, as is typical for reducible sets, projections say nothing about whether the two Minkowski dimensions agree. Nothing suggests that they disagree: for Lebesgue-typical translations the Minkowski dimension exists by Falconer (1988), at least under a norm bound on the linear parts, and in all known reducible examples the upper and lower Minkowski dimensions coincide. The reducible case of the existence question therefore awaits a different mechanism.

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 \(\alpha_2\).