Marstrand's Projection Theorems

• Marstrand’s Projection Theorems are fundamental results in fractal geometry that establish how the Hausdorff dimension of planar sets is preserved under almost every orthogonal projection.
• They demonstrate that for a Borel or analytic set E, the projection onto almost every line retains the minimum of the set’s dimension and 1, with projections having positive Lebesgue measure when the dimension exceeds 1.
• Extensions include generalizations to higher dimensions, applications of potential theory and Fourier analysis, and descriptive-set theoretic counterexamples that clarify the boundaries of these projection laws.

Marstrand’s projection theorems are the foundational statements describing the generic behavior of orthogonal projections of planar sets onto lines. In modern form, if $E\subset \mathbb R^2$ is Borel or analytic and $\operatorname{proj}_\theta E$ denotes orthogonal projection onto the line $L_\theta$ through the origin making angle $\theta$ with the $x$-axis, then for Lebesgue almost every $\theta\in[0,\pi)$,

$$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$

and if $\dim_H E>1$, then the stronger conclusion

$\mathcal L^1(\operatorname{proj}_\theta E)>0$

holds for almost every $\theta$ (Falconer, 25 Feb 2026). These theorems became the prototype for a large part of modern fractal geometry, while later work clarified both their sharpness and the extent to which they persist under changes of dimension notion, projection family, ambient geometry, and definability assumptions (Falconer et al., 2014).

1. Classical planar theorem and its formulation

Marstrand’s original 1954 paper stated the theorem for $\operatorname{proj}_\theta E$0-sets, meaning measurable sets with positive finite $\operatorname{proj}_\theta E$1-dimensional Hausdorff measure. In the formulation recorded in later surveys, the result splits naturally into two regimes. If $\operatorname{proj}_\theta E$2, then for almost every direction,

$\operatorname{proj}_\theta E$3

If $\operatorname{proj}_\theta E$4, then for almost every direction,

$\operatorname{proj}_\theta E$5

which in particular implies

$\operatorname{proj}_\theta E$6

for almost every $\operatorname{proj}_\theta E$7 (Falconer et al., 2014).

The upper bound

$\operatorname{proj}_\theta E$8

is immediate because orthogonal projection is Lipschitz and the target is one-dimensional. The theorem is therefore a reverse inequality for almost every direction. Its content is that typical one-dimensional shadows lose no more dimension than the target forces. The threshold $\operatorname{proj}_\theta E$9 is critical because the image lies in a line: below the threshold, dimension is preserved; above it, the projection saturates at dimension $L_\theta$0, and in fact has positive Lebesgue measure (Falconer, 25 Feb 2026).

The same surveys emphasize that Marstrand’s original paper already contained a stronger subset-uniform statement in the $L_\theta$1 regime: for almost all angles, all positive-$L_\theta$2-measure subsets of a given $L_\theta$3-set have projections of positive length (Falconer et al., 2014). This anticipates later “strong Marstrand theorems.”

2. Potential theory, Fourier analysis, and higher-dimensional extension

A standard modern proof begins from the Frostman-energy characterization

$L_\theta$4

If $L_\theta$5, one chooses $L_\theta$6 with finite $L_\theta$7-energy, projects it to $L_\theta$8, and proves an averaged estimate for projected energies. In Kaufman’s formulation,

$L_\theta$9

so for almost every $\theta$0, the projected measure has finite $\theta$1-energy, implying

$\theta$2

Letting $\theta$3 gives the dimension statement when $\theta$4 (Falconer et al., 2014).

For the positive-measure part, Fourier analysis enters. If $\theta$5 and $\theta$6, a standard argument yields

$\theta$7

Hence for almost every $\theta$8, $\theta$9, so $x$0 is absolutely continuous with an $x$1-density. Its support therefore has positive Lebesgue measure, giving $x$2 (Falconer et al., 2014).

Mattila extended the theorem from planar line projections to orthogonal projections onto $x$3-planes $x$4. For Borel or analytic $x$5,

$x$6

and if $x$7, then

$x$8

(Falconer, 25 Feb 2026).

3. Exceptional directions and strong forms

Later work sharpened the a.e. statement by estimating the exceptional set of bad directions. In the planar case, Kaufman’s classical bound gives

$x$9

when $\theta\in[0,\pi)$0, while Falconer proved

$\theta\in[0,\pi)$1

when $\theta\in[0,\pi)$2 (Falconer et al., 2014). The 2026 survey also records the sharp planar estimate

$\theta\in[0,\pi)$3

together with the statement that the bound is sharp (Falconer, 25 Feb 2026).

A different strengthening concerns uniformity over subsets. If $\theta\in[0,\pi)$4 is $\theta\in[0,\pi)$5-measurable with $\theta\in[0,\pi)$6, Falconer and Mattila proved that there exists a single exceptional set $\theta\in[0,\pi)$7 with $\theta\in[0,\pi)$8 such that for every $\theta\in[0,\pi)$9 and every $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$0-measurable $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$1 with $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$2,

$$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$3

and if $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$4,

$$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$5

They also obtained strong exceptional-set bounds: $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$6 and

$$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$7

(Falconer et al., 2015).

The same paper notes an important limitation: the analogous strong statement for nonempty interior is false. For fixed $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$8, one can remove a countable dense family of fibers perpendicular to $$\dim_H(\operatorname{proj}_\theta E)=\min\{\dim_H E,1\},$$9, leaving a positive-$\dim_H E>1$0-measure subset whose projection has empty interior (Falconer et al., 2015).

4. Refinements below the classical scale and for non-analytic sets

The classical theorem becomes vacuous for sets of ordinary Hausdorff dimension $\dim_H E>1$1, because every projection of such a set also has Hausdorff dimension $\dim_H E>1$2. Beresnevich, Falconer, Velani, and Zafeiropoulos replaced power gauges $\dim_H E>1$3 by finer Hausdorff gauge functions and proved a logarithmic analogue. If $\dim_H E>1$4 is Borel, then for logarithmic Hausdorff dimension,

$\dim_H E>1$5

and

$\dim_H E>1$6

More generally, if $\dim_H E>1$7, $\dim_H E>1$8 is doubling with constant $\dim_H E>1$9, and

$\mathcal L^1(\operatorname{proj}_\theta E)>0$0

then for almost all $\mathcal L^1(\operatorname{proj}_\theta E)>0$1,

$\mathcal L^1(\operatorname{proj}_\theta E)>0$2

(Beresnevich et al., 2017).

A second direction weakens the regularity assumption on the set. Using effective Hausdorff dimension and the point-to-set principle,

$\mathcal L^1(\operatorname{proj}_\theta E)>0$3

Lutz and Stull proved that if $\mathcal L^1(\operatorname{proj}_\theta E)>0$4 satisfies

$\mathcal L^1(\operatorname{proj}_\theta E)>0$5

then for almost every $\mathcal L^1(\operatorname{proj}_\theta E)>0$6,

$\mathcal L^1(\operatorname{proj}_\theta E)>0$7

For arbitrary sets, without analyticity or equality of Hausdorff and packing dimensions, they proved the packing lower bound

$\mathcal L^1(\operatorname{proj}_\theta E)>0$8

(Lutz et al., 2017).

Orponen later gave combinatorial proofs of these Lutz–Stull results and extended them from line projections to projections onto $\mathcal L^1(\operatorname{proj}_\theta E)>0$9-planes. For arbitrary $\theta$0,

$\theta$1

and if

$\theta$2

then

$\theta$3

(Orponen, 2020). In this setting, equality of Hausdorff and packing dimensions functions as a structural replacement for analyticity.

5. Transversality, new geometries, and alternative projection families

A large modern literature treats Marstrand-type theorems as consequences of quantitative transversality. In one abstract formulation, if $\theta$4 are separable metric spaces, $\theta$5 is a probability space, and $\theta$6 is a measurable family satisfying

$\theta$7

then for an analytic $\theta$8,

$\theta$9

and if $\operatorname{proj}_\theta E$00, then typical images have positive $\operatorname{proj}_\theta E$01-dimensional measure (López et al., 2014). A closely related metric-space version with elementary combinatorial methods yields the same dichotomy, together with $\operatorname{proj}_\theta E$02-absolute continuity in the supercritical regime (Moreira et al., 2016).

This transversality viewpoint recovers the Euclidean theorem and extends it to new geometries and projection families. In hyperbolic space, closest-point projection onto totally geodesic $\operatorname{proj}_\theta E$03-planes is conjugate to Euclidean orthogonal projection in the Klein model, and the full Marstrand–Mattila package holds, including exceptional-set bounds and a Besicovitch–Federer theorem (Balogh et al., 2018). On a simply connected surface of non-positive curvature, the positive-measure part survives for projections onto geodesic lines through a fixed point: if $\operatorname{proj}_\theta E$04, then for almost every geodesic line $\operatorname{proj}_\theta E$05,

$\operatorname{proj}_\theta E$06

(Ibarra, 2014).

Norm-induced closest-point projections show that the geometry of the projection family matters independently of bi-Lipschitz equivalence of ambient metrics. In the plane, smooth strictly convex norms with positively curved unit sphere satisfy Marstrand-type and Besicovitch–Federer-type theorems, while norms with corners can fail dramatically (Balogh et al., 2018). In higher dimension, closest-point projections onto hyperplanes in a strictly convex $\operatorname{proj}_\theta E$07-regular normed space satisfy the Euclidean codimension-one conclusions, whereas the paper also constructs a $\operatorname{proj}_\theta E$08-regular norm on $\operatorname{proj}_\theta E$09 for which Marstrand-type theorems fail (Iseli, 2018).

Further extensions replace rotations by larger transformation groups. For linear-fractional families $\operatorname{proj}_\theta E$10 with $\operatorname{proj}_\theta E$11 or $\operatorname{proj}_\theta E$12, local transversality yields Marstrand-type theorems on the natural domains, and restricted one-dimensional subgroup families can also be analyzed in this way (Lukyanenko et al., 2021). Dufloux recast the theory projectively and obtained coordinate-free real and complex versions, including transverse-dimension formulas for foliations by complex chains on spheres (Dufloux, 2017).

A different generalization changes the object being projected. For Borel families of affine lines $\operatorname{proj}_\theta E$13, Gan proved that the almost-everywhere projection dimension onto $\operatorname{proj}_\theta E$14 is governed by an explicit piecewise-linear function $\operatorname{proj}_\theta E$15, rather than the classical $\operatorname{proj}_\theta E$16. This shows that projection laws for families of lines are structurally different from the point case (Gan, 2023).

6. Sharpness, counterexamples, and descriptive-set-theoretic limits

Marstrand’s theorem does not extend to completely arbitrary planar sets. Davies had already shown, under CH, that there exists a set $\operatorname{proj}_\theta E$17 with $\operatorname{proj}_\theta E$18 but all line projections $\operatorname{proj}_\theta E$19-dimensional. Richter sharpened the definability aspect of this phenomenon: assuming $\operatorname{proj}_\theta E$20, there exists a co-analytic set $\operatorname{proj}_\theta E$21 such that

$\operatorname{proj}_\theta E$22

and for every direction $\operatorname{proj}_\theta E$23,

$\operatorname{proj}_\theta E$24

Moreover, for each $\operatorname{proj}_\theta E$25, there is a co-analytic $\operatorname{proj}_\theta E$26 with

$\operatorname{proj}_\theta E$27

such that for every $\operatorname{proj}_\theta E$28,

$\operatorname{proj}_\theta E$29

(Richter, 2023).

These examples are optimal in two senses recorded in the paper. First, a counterexample to Marstrand’s theorem cannot be analytic, because the classical theorem already covers analytic sets. Second, the family $\operatorname{proj}_\theta E$30 attains the general lower bound

$\operatorname{proj}_\theta E$31

so when $\operatorname{proj}_\theta E$32, no construction can force all line projections below $\operatorname{proj}_\theta E$33 (Richter, 2023).

The proof of these co-analytic counterexamples differs sharply from classical potential theory. It combines descriptive set theory, transfinite recursion under $\operatorname{proj}_\theta E$34, and algorithmic dimension via the point-to-set principle. The resulting picture is that, under $\operatorname{proj}_\theta E$35, analytic sets still obey Marstrand’s theorem, while co-analytic sets can fail it maximally. This locates a sharp descriptive-set-theoretic boundary for the theorem’s validity (Richter, 2023).

Taken together, the classical theorem, its potential-theoretic proofs, the strong and exceptional-set refinements, the extensions to generalized projection families, and the co-analytic counterexamples show that Marstrand’s projection theorems are both robust and genuinely sensitive to structure. They remain central because they identify a generic projection law, quantify its failures, and provide a template for projection theory across fractal geometry, geometric measure theory, and related parts of analysis (Falconer, 25 Feb 2026).