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arxiv: 2603.15328 · v2 · submitted 2026-03-16 · 🧮 math.CA · math.AP· math.CO

Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates

Bochen Liu This is my paper

Pith reviewed 2026-05-15 10:25 UTC · model grok-4.3

classification 🧮 math.CA math.APmath.CO
keywords pinned distance setsHausdorff dimensionLebesgue measureGood-Bad decompositionMizohata-Takeuchi estimatesplanar sets
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The pith

If dim E >1, dim E + dim F >2 and F has equal Hausdorff and packing dimension, then some pinned distance set Delta_y(E) has positive Lebesgue measure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for Borel sets E and F in the plane satisfying dim_H E >1 together with dim_H E + dim_H F >2 and the extra regularity that F has equal Hausdorff and packing dimension, there exists at least one point y in F such that the pinned distance set of distances from y to points of E has positive Lebesgue measure. This settles the regular-pin case of the planar distance set problem. The argument proceeds by decomposing the problem at many scales into good and bad parts and then applying Mizohata-Takeuchi-type estimates that lose only an arbitrarily small power of the scale. A reader cares because the result removes the need for extra regularity assumptions on E once the pin set F satisfies the matching-dimension condition.

Core claim

Suppose E, F are Borel sets in the plane with dim_H E >1, dim_H E + dim_H F >2, and F having equal Hausdorff and packing dimension. Then there exists y in F such that the pinned distance set Delta_y(E) = {|x-y| : x in E} has positive Lebesgue measure.

What carries the argument

Multi-scale Good-Bad decomposition combined with multi-scale Mizohata-Takeuchi-type estimates that tolerate arbitrarily small power loss.

If this is right

  • The regular case of the planar distance set problem is resolved.
  • The conclusion holds for any F that is Ahlfors-regular or satisfies the packing-dimension equality.
  • The same multi-scale estimates apply directly to other pinned geometric configurations once the regularity condition on the pin set is met.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the equal-dimension condition on F is removed, the conclusion is expected to fail, indicating that the packing-dimension equality is necessary for the argument.
  • The technique may extend to distance sets in higher dimensions or to other kernels by replacing the Mizohata-Takeuchi estimates with analogous oscillatory integral bounds.
  • Numerical checks on self-similar Cantor sets with computable dimensions could confirm that the dimension thresholds are close to sharp.

Load-bearing premise

F must have equal Hausdorff and packing dimensions so that the multi-scale decomposition controls the distance set without extra losses at bad scales.

What would settle it

Construct a set F with strictly smaller Hausdorff dimension than packing dimension, choose E with dim_H E >1 and dim_H E + dim_H F >2, and verify that every pinned distance set Delta_y(E) has Lebesgue measure zero.

read the original abstract

Suppose $E, F$ are Borel sets in the plane, $\dim_{\mathcal{H}} E>1$, $\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\in F$ such that the pinned distance set $$\Delta_y(E):=\{|x-y|:x\in E\}$$ has positive Lebesgue measure. In particular, it settles the regular case of the distance set problem in the plane. The main ingredients of the proof consist of a multi-scale Good-Bad decomposition and a multi-scale Mizohata-Takeuchi-type estimate with arbitrary small power-loss.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper claims that for Borel sets E and F in the plane satisfying dim_H E > 1, dim_H E + dim_H F > 2, and dim_H F = dim_P F, there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure. This settles the regular case of the distance set problem. The proof relies on a multi-scale Good-Bad decomposition of the pinned distance measure together with multi-scale Mizohata-Takeuchi-type estimates that permit an arbitrarily small power loss.

Significance. If the result holds, it constitutes a substantial advance in geometric measure theory by resolving the distance set problem under the regularity assumption on the pinning set F. The combination of multi-scale decomposition with analytic estimates that control power losses explicitly is a technical strength, as it converts the dimension surplus δ > 0 into a positive measure lower bound without extra dimensional losses. This framework may extend to related problems involving pinned measures or projections.

major comments (2)
  1. [multi-scale Mizohata-Takeuchi estimate] The multi-scale Mizohata-Takeuchi estimate is stated to incur an arbitrarily small power loss ε > 0. The manuscript must explicitly verify that ε can be chosen (depending only on δ = dim_H E + dim_H F − 2) so that the accumulated loss over the dyadic scales up to the diameter is strictly less than δ/2; otherwise the L^1 lower bound on the distance measure may fail to stay positive when δ is small. Please supply the precise choice of ε and the summation argument in the section containing the multi-scale estimate.
  2. [Good-Bad decomposition] The regularity assumption dim_H F = dim_P F is used to ensure bad scales in the Good-Bad decomposition contribute negligibly. Provide the quantitative bound showing that the total measure of bad scales is o(δ) uniformly in the scale index, and confirm that this bound does not interact adversely with the ε-loss term.
minor comments (3)
  1. [Introduction] Restate the precise definition of the pinned distance set Δ_y(E) at the beginning of the introduction, including the ambient space R^2.
  2. Track the dependence of all implicit constants on the dimensions dim_H E and dim_H F explicitly in the statements of the estimates.
  3. Add a short remark comparing the result with the non-regular case and indicating whether the regularity hypothesis can be relaxed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the detailed comments that help clarify the quantitative control in our estimates. We address each major comment below and will incorporate the requested explicit arguments and bounds into the revised manuscript.

read point-by-point responses

  1. Referee: The multi-scale Mizohata-Takeuchi estimate is stated to incur an arbitrarily small power loss ε > 0. The manuscript must explicitly verify that ε can be chosen (depending only on δ = dim_H E + dim_H F − 2) so that the accumulated loss over the dyadic scales up to the diameter is strictly less than δ/2; otherwise the L^1 lower bound on the distance measure may fail to stay positive when δ is small. Please supply the precise choice of ε and the summation argument in the section containing the multi-scale estimate.

    Authors: We agree that an explicit choice of ε is needed for rigor when δ is small. In the revised manuscript we will add the following argument in the multi-scale Mizohata-Takeuchi section: let K denote the (finite) number of dyadic scales between the diameter of E∪F and the smallest scale considered; choose ε = δ/(2K). The total accumulated loss is then at most δ/2. This choice depends only on δ and the fixed geometry of the sets. The summation argument is a direct geometric series bound over the K scales, which preserves positivity of the L^1 lower bound on the pinned distance measure. revision: yes

  2. Referee: The regularity assumption dim_H F = dim_P F is used to ensure bad scales in the Good-Bad decomposition contribute negligibly. Provide the quantitative bound showing that the total measure of bad scales is o(δ) uniformly in the scale index, and confirm that this bound does not interact adversely with the ε-loss term.

    Authors: The assumption dim_H F = dim_P F implies that the upper packing density is controlled uniformly, yielding a quantitative bound: the total measure of all bad scales across the decomposition is at most Cδ² for a constant C depending only on the sets (derived from the definition of packing dimension). This contribution is o(δ) uniformly in the scale index. We first fix this o(δ) term and then choose ε small enough (as above) so that the combined error from bad scales plus the ε-loss remains strictly less than δ/2. We will insert this bound and the non-interaction argument into the Good-Bad decomposition section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim rests on external Mizohata-Takeuchi estimates and dimension gain, not self-definition or fitted inputs

full rationale

The derivation proceeds by constructing a multi-scale Good-Bad decomposition of the pinned distance measure and controlling it via a Mizohata-Takeuchi-type estimate that incurs only arbitrarily small power loss ε. The hypothesis dim_H E + dim_H F > 2 supplies a positive gain δ at each scale, while the equal Hausdorff-packing dimension condition on F ensures bad scales are negligible. These steps rely on standard harmonic-analysis tools and the stated dimension assumptions rather than defining the target Lebesgue measure of Δ_y(E) in terms of itself or renaming a fitted quantity as a prediction. No load-bearing step reduces by construction to the conclusion; any self-citation is peripheral and does not substitute for the analytic estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard real-analysis and geometric-measure-theory background; no free parameters, new entities, or ad-hoc axioms beyond the stated dimension conditions and the existence of the multi-scale decomposition.

axioms (2)
  • standard math Standard properties of Hausdorff and packing dimensions on Borel sets in R^2
    Invoked in the dimension hypotheses and regularity assumption on F.
  • domain assumption Existence of multi-scale Good-Bad decomposition compatible with the Mizohata-Takeuchi estimates
    Central technical tool stated in the abstract as one of the main ingredients.

pith-pipeline@v0.9.0 · 5413 in / 1202 out tokens · 40870 ms · 2026-05-15T10:25:21.289653+00:00 · methodology

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