Pinned Distance Sets

• Pinned distance sets are one-point variants of classical distance sets, examining all distances from a fixed point in a metric space.
• They play a critical role in geometric measure theory, harmonic analysis, and additive combinatorics by analyzing dimensions and density measures.
• Recent advances explore density, fractal, and algebraic settings, revealing a structural dichotomy between global and pinned configurations.

Pinned distance sets are the one-point variants of classical distance sets: for a set $E$ in a metric space and a fixed pin $x$, one studies the set of distances from $x$ to points of $E$. In Euclidean space this is typically written as

$$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$

while the unpinned distance set is $D(A)=\{|x-y|:x,y\in A\}$ (Wang, 1 Sep 2025). The subject lies at the intersection of geometric measure theory, harmonic analysis, additive combinatorics, and incidence geometry. Its main questions concern the size of $D_x(E)$ or $\Delta_x(E)$, measured by upper density, Hausdorff dimension, box-counting dimension, Lebesgue measure, or interior, and how these depend on the largeness assumptions imposed on $E$: positive upper density, Hausdorff dimension $>1$, Ahlfors regularity, Fourier decay, or combinatorial cardinality over finite algebraic structures.

1. Definitions, ambient settings, and principal notions

In the Euclidean setting, the basic definitions are the distance set

$x$0

and the pinned distance set

$x$1

for measurable $x$2 (Wang, 1 Sep 2025). For sets of positive density, the relevant largeness parameter is the upper density

$x$3

where $x$4 denotes Lebesgue measure (Wang, 1 Sep 2025). For fractal sets, the main size parameters are Hausdorff dimension, packing dimension, lower box-counting dimension, and modified lower box-counting dimension (Shmerkin, 2018, Shmerkin, 2016).

The notation varies across the literature. In many planar and higher-dimensional fractal papers the pinned distance set is written as

$x$5

or $x$6 (Shmerkin, 2017, Iosevich et al., 2017). In Riemannian settings, if $x$7 is the geodesic metric on a compact manifold $x$8, the pinned set is

$x$9

(Iosevich et al., 2016). In finite fields and finite valuation rings one replaces Euclidean distance by an algebraic quadratic form such as

$x$0

and studies the cardinality of the pinned distance set at a pin $x$1 or $x$2 (Yazici, 2017, Koh, 2022).

A recurring structural distinction is between statements about the global distance set and statements about a single pin. Several recent results show that the pinned problem is not merely a localized reformulation of the unpinned one. This distinction is especially visible in positive-density problems, where global interval-filling conclusions can fail after pinning even when quantitative largeness survives (Wang, 1 Sep 2025).

2. Positive upper density and the failure of a Bourgain-type pinned theorem

For measurable sets of positive upper density in $x$3, Bourgain’s theorem asserts that there exists $x$4 such that $x$5, so all sufficiently large distances are realized globally (Wang, 1 Sep 2025). A natural pinned analogue asks whether, for $x$6 of positive upper density, there exists a single point $x$7 such that every sufficiently large distance is realized as $x$8 with $x$9.

That analogue fails. The paper "Pinned distances and density theorems in $E$0" proves that there exists a set $E$1 with positive upper density such that for every $E$2 and every $E$3, there is $E$4 for which no $E$5 satisfies $E$6 (Wang, 1 Sep 2025). Equivalently, for every $E$7, $E$8 fails to contain any interval of the form $E$9. The construction is a union of widely separated large cubes whose distances from one another grow so fast that, for each point $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$0, the attainable distances have arbitrarily large gaps (Wang, 1 Sep 2025).

The same work establishes a weaker quantitative replacement. If $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$1 has $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$2, then for every $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$3,

$$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$4

Thus every pinned distance set has positive upper density in $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$5, with a lower bound linear in $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$6 (Wang, 1 Sep 2025). The proof uses spherical coordinates, radial integration of the characteristic function of $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$7, and translation invariance of upper density (Wang, 1 Sep 2025).

This quantitative statement is sharp up to constant factors. A sparse union of thin annuli can have positive upper density while, for every $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$8, the pinned distance set has upper density at most $$D_x(A):=\{|x-y|:y\in A\} \quad\text{or}\quad \Delta_x(E):=\{|x-y|:y\in E\},$$9 (Wang, 1 Sep 2025). A plausible implication is that positive upper density controls radial occupation only in an averaged-density sense, not through interval structure. This is one of the clearest known manifestations of a structural dichotomy between global and pinned configurations in Euclidean distance problems (Wang, 1 Sep 2025).

Related dense-set analogues also occur in $D(A)=\{|x-y|:x,y\in A\}$0. For $D(A)=\{|x-y|:x,y\in A\}$1 with positive upper Banach density and $D(A)=\{|x-y|:x,y\in A\}$2, discrete spherical maximal function methods yield pinned variants asserting that there is a fixed $D(A)=\{|x-y|:x,y\in A\}$3 from which many large discrete radii are realized uniformly over a range of $D(A)=\{|x-y|:x,y\in A\}$4 (Lyall et al., 2015). This suggests that the Euclidean positive-density obstruction in the pinned setting is specific to the continuous upper-density framework addressed in (Wang, 1 Sep 2025), rather than a universal feature of all dense-set models.

3. Planar fractal sets above dimension $D(A)=\{|x-y|:x,y\in A\}$5

For planar Borel or analytic sets with Hausdorff dimension $D(A)=\{|x-y|:x,y\in A\}$6, one asks whether pinned distance sets are large in Hausdorff dimension, box dimension, or measure. A landmark result states that if $D(A)=\{|x-y|:x,y\in A\}$7 is Borel with

$D(A)=\{|x-y|:x,y\in A\}$8

then

$D(A)=\{|x-y|:x,y\in A\}$9

In particular, for all $D_x(E)$0 outside a set of Hausdorff dimension at most $D_x(E)$1, the pinned distance set has full Hausdorff dimension $D_x(E)$2 (Shmerkin, 2017). The proof combines multi-scale decomposition with hyperdyadic scales, energy estimates, entropy methods, and quantitative circular projection theorems (Shmerkin, 2017).

Subsequent work sharpened explicit lower bounds when only $D_x(E)$3 is assumed. Keleti and Shmerkin showed that if $D_x(E)$4 is Borel with $D_x(E)$5, then outside an exceptional set of dimension at most $D_x(E)$6,

$D_x(E)$7

and for $D_x(E)$8,

$D_x(E)$9

(Keleti et al., 2018). Their method introduced a multi-scale decomposition with flexible scales and a combinatorial optimization problem on variations of Lipschitz functions (Keleti et al., 2018).

Shmerkin later improved the planar pinned Hausdorff-dimension bound for sets with $\Delta_x(E)$0. For all $\Delta_x(E)$1 outside a set of Hausdorff dimension at most $\Delta_x(E)$2,

$\Delta_x(E)$3

and if $\Delta_x(E)$4, there are many $\Delta_x(E)$5 such that

$\Delta_x(E)$6

where $\Delta_x(E)$7 for $\Delta_x(E)$8 (Shmerkin, 2018). The argument builds on the Keleti–Shmerkin framework and incorporates estimates from Guth, Iosevich, Ou, and Wang, together with spherical projection theorems of Orponen (Shmerkin, 2018).

Algorithmic methods led to further bounds. Using effective dimension and Kolmogorov complexity, Stull proved that for any analytic $\Delta_x(E)$9 with $E$0,

$E$1

for all points $E$2 outside a set of Hausdorff dimension at most $E$3 (Stull, 2022). For $E$4 close to $E$5, this improves the then-best known explicit lower bounds (Stull, 2022).

Semi-regularity assumptions yield finer interpolation between Hausdorff and packing dimensions. If $E$6 is analytic with $E$7 and $E$8, then for all $E$9 in a subset of full Hausdorff dimension,

$>1$0

and if

$>1$1

then $>1$2 (Fiedler et al., 2023). This makes the regularity gap $>1$3 an explicit parameter in the pinned problem.

A common misconception is that $>1$4 alone already forces full-dimensional pinned distance sets for most pins without further hypotheses. The literature shows a more stratified picture: full Hausdorff dimension is known under equal Hausdorff and packing dimension (Shmerkin, 2017), while under weaker assumptions one obtains explicit lower bounds that depend on $>1$5, $>1$6, or related regularity parameters (Shmerkin, 2018, Stull, 2022, Fiedler et al., 2023).

4. Low-dimensional planar sets, regularity, and universal pin sets

The planar pinned distance problem below the threshold $>1$7 has developed along two directions: lower bounds for $>1$8, and energy inequalities that extend the pinned theory into the $>1$9 regime.

For analytic $x$00 with $x$01 and $x$02, one has

$x$03

where $x$04 (Fiedler et al., 2024). In the special case $x$05,

$x$06

(Fiedler et al., 2024). The bound improves as the set becomes more regular, in the sense that $x$07 and $x$08 become closer (Fiedler et al., 2024).

The same work identifies “weakly universal” and universal pin sets. If $x$09 is weakly regular, meaning $x$10, then for every Borel set $x$11,

$x$12

If $x$13 is also compact and Ahlfors-David regular, then for every Borel set $x$14, there exists $x$15 such that

$x$16

(Fiedler et al., 2024). The paper gives the 4-corner Cantor set with $x$17 as an example of such a universal pin set (Fiedler et al., 2024).

Low-dimensional pinned distance sets are also accessible via spherical averages. Harris derived an inequality for the average $x$18-energy of pinned distance measures for $x$19, refining Mattila’s theorem to the pinned setting: $x$20 under the condition

$x$21

(Harris, 2021). This provides an analogue of Liu’s theorem for pinned distance sets of dimension smaller than $x$22 and opens a genuinely low-dimensional pinned regime that earlier approaches did not address (Harris, 2021).

Regularity is also decisive in box-counting formulations. For bounded subsets of planar $x$23-Ahlfors regular sets with $x$24, Shmerkin proved that for $x$25-almost all $x$26,

$x$27

and gave exceptional set estimates for pins whose lower box-counting dimension falls below a given $x$28 (Shmerkin, 2016). The proofs use CP-processes, entropy for projections, and ergodic-theoretic scaling scenery (Shmerkin, 2016).

These results suggest that “regularity” in the pinned problem is not a secondary hypothesis. Depending on context, it may mean equality of Hausdorff and packing dimensions, Ahlfors-David regularity, or suitable multiscale uniformity, and each form of regularity strengthens what can be said about a single pin (Fiedler et al., 2024, Shmerkin, 2016).

5. Positive Lebesgue measure, higher dimensions, and analytic frameworks

Beyond dimension lower bounds, a central objective is to prove that $x$29 has positive Lebesgue measure, or even non-empty interior. A general higher-dimensional result states that for Borel sets $x$30, there exists a probability measure $x$31 on $x$32 such that for $x$33-almost every $x$34,

$x$35

$x$36

and

$x$37

(Iosevich et al., 2017). The proof uses local smoothing estimates for Fourier integral operators, Frostman measures, and slicing arguments (Iosevich et al., 2017).

A variable-coefficient and manifold version was established for compact $x$38-dimensional Riemannian manifolds without boundary. If $x$39 has Hausdorff dimension greater than $x$40, then there are many $x$41 such that the Lebesgue measure of $x$42 is positive, and the bad set satisfies

$x$43

(Iosevich et al., 2016). The argument is based on Radon transform estimates under smoothness, non-degeneracy, and Monge–Ampère determinant conditions (Iosevich et al., 2016).

In higher-dimensional Euclidean space, Du, Ou, Ren, and Zhang improved the Lebesgue-positivity threshold. If $x$44 is compact and $x$45, then

$x$46

(Du et al., 2023). They also proved Hausdorff-dimension lower bounds for pinned distance sets in an intermediate range, using

$x$47

to obtain $x$48 under the stated dimensional hypotheses (Du et al., 2023). The method introduces a dichotomy involving heavy plates, a new radial projection theorem, and refined decoupling (Du et al., 2023).

For planar sets with regular pins, a later result proves a Lebesgue-positivity statement at the conjectural threshold in the regular case. If $x$49 are Borel, $x$50, $x$51, and $x$52 has equal Hausdorff and packing dimension, then there exists $x$53 such that $x$54 (Liu, 16 Mar 2026). The proof uses a multi-scale Good-Bad decomposition together with a multi-scale Mizohata-Takeuchi-type estimate with arbitrarily small power loss (Liu, 16 Mar 2026). In the terminology of that paper, this settles the regular case of the distance set problem in the plane (Liu, 16 Mar 2026).

Fourier-analytic refinements yield another route. Under Fourier spectrum assumptions, one can bound the Hausdorff dimension of typical pinned distance sets. If $x$55 are compactly supported probability measures with Fourier-spectrum data $x$56 and $x$57, then for $x$58-almost every $x$59,

$x$60

with $x$61 given piecewise in terms of $x$62, and if $x$63, then the pinned distance set has positive Lebesgue measure (Fraser et al., 21 Apr 2026). In particular, if $x$64, then for $x$65-almost all $x$66,

$x$67

(Fraser et al., 21 Apr 2026). The paper also constructs sharpness or near-sharpness examples (Fraser et al., 21 Apr 2026).

A recurring theme across these analytic approaches is that the output depends strongly on the type of largeness assumed. Hausdorff dimension, regularity, local smoothing input, and Fourier decay each support different pinned conclusions: lower dimension bounds, positive measure, or full dimension for almost every pin (Iosevich et al., 2017, Du et al., 2023, Liu, 16 Mar 2026, Fraser et al., 21 Apr 2026).

6. Product structures, algebraic settings, and discrete analogues

Pinned distance problems admit strong algebraic and combinatorial formulations over Cartesian products, finite fields, finite valuation rings, and dense subsets of lattices.

For Cartesian product sets in Euclidean space, the parabolic method gives improved thresholds. If $x$68 are compact and $x$69, then under explicit dimensional conditions involving $x$70, $x$71, and $x$72, there exists $x$73 such that

$x$74

and related threshold statements yield interval containment or positive Lebesgue measure for the pinned distance set (Li et al., 18 Mar 2025). The method replaces Euclidean distance by a parabolic distance

$x$75

to exploit Phong–Stein and cinematic curvature in combination with sharp planar pinned estimates (Li et al., 18 Mar 2025).

Over finite valuation rings, if $x$76 has order $x$77 with $x$78 odd and $x$79, then there exists $x$80 such that

$x$81

In particular, if $x$82, then $x$83 determines a positive proportion of all possible distances from a single pin (Yazici, 2017). The proof passes through point-plane incidences in $x$84 (Yazici, 2017).

Over fields of positive characteristic, pinned distance results exhibit regime changes by cardinality. If $x$85 with $x$86, then $x$87, where $x$88 is the maximum pinned distance count over pins $x$89; moreover, for sufficiently large $x$90, if $x$91, then for at least $x$92 points $x$93,

$x$94

(Murphy et al., 2020). If $x$95, then either $x$96 lies in a single isotropic line or $x$97 (Murphy et al., 2020). The arguments use bisector energy, incidence geometry, and the Blaschke–Grünwald mapping (Murphy et al., 2020).

A complementary finite-field result shows that if $x$98 with $x$99, then there exists $x$00 with $x$01 such that for all $x$02, the number of pinned distances between $x$03 and $x$04 is comparable to $x$05 (Koh, 2022). More generally, for any $x$06, there exists $x$07 with $x$08 such that for all $x$09,

$x$10

(Koh, 2022). The proof is based on averaging and the pigeonhole principle (Koh, 2022).

Generalized pinned distance problems over finite fields replace the quadratic form by a polynomial $x$11. Writing

$x$12

one obtains large pinned distance sets for diagonal and other polynomial distance functions under Fourier decay assumptions on the level sets $x$13 (Koh et al., 2010). This extends spherical and cubic distance problems to a broader algebraic class (Koh et al., 2010).

Finally, in dense subsets of $x$14 with $x$15, discrete spherical averages and maximal theorems produce pinned variants in which a single $x$16 realizes many large radii $x$17 with density close to $x$18 over a whole range of $x$19 (Lyall et al., 2015). These results parallel continuous dense-set theorems while revealing arithmetic obstructions absent from the Euclidean continuum (Lyall et al., 2015).

Pinned distance sets therefore form a family of problems rather than a single theorem. In Euclidean positive-density settings they exhibit a sharp global-versus-pinned separation (Wang, 1 Sep 2025); in planar fractal geometry they are strongly sensitive to regularity, exceptional-set size, and the distinction between Hausdorff, packing, and box dimensions (Shmerkin, 2017, Shmerkin, 2018, Shmerkin, 2016); in higher dimensions they are closely tied to local smoothing, projection theory, decoupling, and Fourier decay (Iosevich et al., 2017, Du et al., 2023, Fraser et al., 21 Apr 2026); and in algebraic or discrete settings they connect to incidence geometry, energy methods, and polynomial Fourier analysis (Yazici, 2017, Murphy et al., 2020, Koh et al., 2010). The cumulative picture is that pinning exposes geometric and analytic constraints that are often invisible in the global distance set problem.