Falconer Distance Conjecture

Falconer Distance Conjecture

OpenAI's claim

Falconer distances: For every $d\ge2$, the manuscript claims positive one-dimensional measure for the distance set of a compact subset of $\mathbb R^d$ with dimension exceeding $d/2$. Both the ambient dimension and the set itself are unrestricted within these hypotheses.

Mathematical statement of the claimed result

Mathematical claim 1.

For every integer $d\ge2$ and compact $E\subset\mathbb R^d$, \[\dim_H E>d/2\quad\Longrightarrow\quad\mathcal L^1\left(\{|x-y|:x,y\in E\}\right)>0.\] Here $\dim_H$ is Hausdorff dimension and $\mathcal L^1$ is Lebesgue measure on the distance axis.

Outline of the argument

Falconer distances: Auxiliary projections first provide angular nonconcentration and filters on pairs of points. A fractal Fourier estimate and a graph estimate with separate angular variables control shell approximations after small additional deletions. The filters are brought back to distances of the original pairs, and monotone shell summation produces a nonzero absolutely continuous distance measure.

Formal verification

Family 073 describes Lean coverage for the positive-measure distance conclusion above the half-dimensional threshold.

Falconer distances: The manuscript links FalconerAllDimensions.lean, PlanarFalconer.lean (declaration OAI.PlanarFalconer.mainTarget_proved).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

• OpenAI, The Falconer distance conjecture in all dimensions, 2026-09-23; citation key OAI:The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 073: Lean scope.
• Comparator FalconerAllDimensions.json.
• Comparator FalconerAllDimensions.lean.
• Comparator PlanarFalconer.json.
• Comparator PlanarFalconer.lean.
• OpenAI catalogue entry.

Falconer Distance Conjecture

Definitions and mathematical statement

For compact $E\subset{\mathbb{R}}^d$, define its Euclidean distance set $\Delta(E)=\{|x-y|:x,y\in E\}\subset[0,\infty)$. Hausdorff dimension is $\dim_H E=\inf\{s:\mathcal H^s(E)=0\}$, with $\mathcal H^s$ defined by arbitrarily fine covers and sums of $s$th powers of diameters. The conjecture is \[ \dim_H E>d/2\quad\Longrightarrow\quad \mathcal L^1(\Delta(E))>0 \qquad(d\ge2). \] The threshold inequality is strict. Positive one-dimensional measure is stronger than merely uncountably many distances or a distance set of Hausdorff dimension one. The set need not carry positive $d$-dimensional volume.

Formulation and concept references: [S1].

Short English statement

Must a compact Euclidean set whose Hausdorff dimension exceeds half the ambient dimension determine a positive-measure set of distances?

Research attempt

Current frontier.

The planar theorem of Guth--Iosevich--Ou--Wang reaches dimension $>5/4$, with a pinned positive-measure conclusion.[E1] Du--Ou--Ren--Zhang, Theorem 1.1 in the 2024 revision, obtain the threshold $d/2+1/4-1/(8d+4)$ for $d\ge3$, again with a pinned conclusion. [E2] These exceed $d/2$. Their use of carefully selected or modified measures is important: a conjectural estimate for every measure on $E$ would be stronger than the set-existence statement. The attempt below only asks to find one suitable probability measure.

Attack route.

Fourier $L^2$ estimates give a strong way to force positive measure, but may pay too much for rare, highly populated distance annuli. I investigate an entropy budget instead. It penalizes annular imbalance logarithmically and has an exact nonnegative increment at each dyadic refinement. The possible advantage is that one can tolerate some large collision energies; the missing lemma is summability of the actual distance-measure imbalance, not merely a dimension-one conclusion. A sparse-digit construction tests whether a weaker asymptotic entropy statement could suffice.

Attempt.

1. Normalize the distance measure without changing the target. Fix compact $E$ with $\dim_H E>d/2$ and choose $L>\operatorname{diam}(E)$. For a Borel probability measure $\mu$ supported on $E$, let \[ \nu(A)=(\mu\times\mu)\{(x,y): |x-y|/L\in A\},\qquad A\subset[0,1). \] Its support lies in $\Delta(E)/L$, a compact set. The measure need not have full support on $E$; any positive-measure subset of the distance set suffices. Let $I_{j,k}=[k2^{-j},(k+1)2^{-j})$, $0\le k<2^j$, and define \[ p_{j,k}=\nu(I_{j,k}),\qquad H_j=-\sum_k p_{j,k}\log p_{j,k},\qquad D_j=j\log2-H_j, \] with $0\log0=0$ and natural logarithms. The normalization $L>\operatorname{diam}(E)$ avoids an endpoint mass at $1$.

2. Bounded entropy deficit forces positive measure. For completeness, a direct support argument avoids assuming a density for $\nu$. If $N_j$ cells have positive mass, concavity of the logarithm gives $H_j\le\log N_j$, and hence \begin{equation}\label{wsref-0-33-30-30-30-31-30-36-33-3a-73-75-70-70-6f-72-74} 2^{-j}N_j\ge e^{-D_j}. \end{equation} Let $U_j$ be the union of the closed cells corresponding to positive masses. The sets $U_j$ are compact and nested: a positive-mass child has a positive-mass parent. Their intersection is contained in $\operatorname{supp}\nu$. Indeed, a point outside that support has a neighborhood of zero mass and cannot belong to such a cell once the cell diameter is sufficiently small. Closing cells changes no Lebesgue measure. If $D_j\le C$ for every $j$, continuity of Lebesgue measure from above and (30001063:support) yield \begin{equation}\label{wsref-0-33-30-30-30-31-30-36-33-3a-70-6f-73-69-74-69-76-65} |\Delta(E)|\ge L|\operatorname{supp}\nu|\ge Le^{-C}>0. \end{equation} There is no assumption here that Hausdorff dimension one already implies positive measure.

3. An exact annular-imbalance identity. For a parent of positive mass, put \[ \beta_{j,k}=\frac{p_{j+1,2k}-p_{j+1,2k+1}}{p_{j,k}}\in[-1,1], \qquad \Phi(b)=\frac{(1+b)\log(1+b)+(1-b)\log(1-b)}2. \] Set the summand below to zero when $p_{j,k}=0$. Expanding the entropy of both children gives the exact chain rule \begin{equation}\label{wsref-0-33-30-30-30-31-30-36-33-3a-63-68-61-69-6e} D_{j+1}-D_j=\sum_k p_{j,k}\Phi(\beta_{j,k})=:B_j\ge0, \qquad D_0=0. \end{equation} Here $\Phi(0)=0$, $\Phi(\pm1)=\log2$, and $\Phi''(b)=1/(1-b^2)$ on $(-1,1)$. Thus $\Phi(b)\ge b^2/2$, while $\Phi(b)\le(2/3)b^2$ for $|b|\le1/2$. All endpoint values are limits, so a parent whose entire mass chooses one child is included rather than silently excluded.

This proves the following sufficient criterion, including its constants. Define \[ a_j=\sum_{k:\,|\beta_{j,k}|>1/2}p_{j,k},\qquad b_j=\sum_{k:\,|\beta_{j,k}|\le1/2}p_{j,k}\beta_{j,k}^2. \] If $\sum_j a_j<\infty$ and $\sum_j b_j<\infty$, then \begin{equation}\label{wsref-0-33-30-30-30-31-30-36-33-3a-62-75-64-67-65-74} |\Delta(E)|\ge L\exp\left(-\log2\sum_j a_j-\frac23\sum_j b_j\right)>0. \end{equation} The proof is just the upper bounds for $\Phi$, telescoping (30001063:chain), and (30001063:positive). This allows very biased cells provided their total probability mass, summed over scales, is finite.

4. What the missing geometric estimate actually says. In terms of the original points, define the alternating annular indicator \[ A_{j,k}(t)=\mathbf1_{I_{j+1,2k}}(t)-\mathbf1_{I_{j+1,2k+1}}(t). \] Then the difference tested in the imbalance is exactly \[ p_{j+1,2k}-p_{j+1,2k+1} =\iint_E A_{j,k}(|x-y|/L)\,d\mu(x)d\mu(y). \] A completion would establish, for every admissible $E$, the existence of $\mu$ for which the sum of the $B_j$ in (30001063:chain) is finite. This is a sufficient condition, not an asserted equivalent reformulation of Falconer: a positive-measure distance set need not be witnessed by a measure with this entropy bound.

For comparison let $\mathcal E_j=2^j\sum_kp_{j,k}^2$. Jensen's inequality with weights $p_{j,k}$ gives \[ D_j=\sum_kp_{j,k}\log(2^jp_{j,k})\le\log\mathcal E_j. \] Uniform dyadic $L^2$ control therefore implies the entropy criterion. The converse is false even for ordinary densities: an integrable probability density proportional near zero to $x^{-1/2}(\log(e/x))^{-1/4}$ has finite $\int f\log f$ but infinite $\int f^2$. Its dyadic deficits are bounded by $\int f\log f$ through cellwise Jensen. This demonstrates a real analytic weakening, although it does not establish that a geometric construction can exploit it.

5. A sharp warning about sublinear losses. Take independent binary digits $\xi_r$, except that $\xi_r=0$ whenever $r$ is a perfect square, and put \[ X=\sum_{r\ge1}\xi_r2^{-r}; \] all other digits are fair. Let $F$ be the compact set of all such expansions and $\sigma$ the resulting probability law. Ambiguous dyadic expansions form a countable set of $\sigma$-measure zero, because there are infinitely many free digits. At level $j$ exactly $2^{j-\lfloor\sqrt j\rfloor}$ cells have positive mass, all equal. Consequently \begin{equation}\label{wsref-0-33-30-30-30-31-30-36-33-3a-73-70-61-72-73-65} H_j(\sigma)=(j-\lfloor\sqrt j\rfloor)\log2, \quad D_j(\sigma)=\lfloor\sqrt j\rfloor\log2, \quad \mathcal E_j(\sigma)=2^{\lfloor\sqrt j\rfloor}. \end{equation} These occupied cells cover $F$ up to their endpoints and have total length tending to zero, so $|F|=0$. Nevertheless $\dim_H F=1$. To verify the lower bound, for every $s<1$ the largest level-$j$ cell mass is at most $C_s2^{-sj}$. An interval of length between $2^{-j-1}$ and $2^{-j}$ meets at most two level-$j$ cells apart from null endpoints. Hence $\sigma(I)\le C'_s|I|^s$, which implies $\dim_H F\ge s$ by applying this inequality to arbitrary small covers. Let $s\uparrow1$.

This example has $D_j=o(j)$ and $\mathcal E_j=2^{o(j)}$, yet zero measure. Its entropy cost is concentrated at the sparse forced-digit scales: $B_{r-1}=\log2$ for square $r$ and zero otherwise. An arbitrarily long run of balanced scales gives no control of the total future deficit.

Stress test.

The sparse-digit example is a measure on the line, not a constructed counterexample distance set of a high-dimensional $E$. Its role is to falsify the inference from sublinear entropy loss to positive measure. The compact-support proof handles dyadic boundaries explicitly and does not interchange an unproved density limit with integration. The chain rule is finite at each scale and telescopes monotonically, so its use requires no cancellation of a conditionally convergent series.

The central unproved claim is the existence of a measure satisfying the summable annular-imbalance budget for every compact set of dimension $>d/2$. Frostman ball bounds alone do not furnish this: annuli constrain pairs and their alternating differences, not just mass inside balls. Known pinned-measure modifications should not be assumed to preserve this new budget without a separate proof.[E1][E2]

\Needspace{11\baselineskip}

Outcome.

Outcome: Conditional reduction.

A summable entropy-imbalance estimate implies the exact positive-measure conclusion, with the explicit lower bound (30001063:budget). The proof also separates this estimate from both uniform $L^2$ control and a mere full-dimensional distance conclusion. No summable budget has been proved under the conjecture's hypotheses alone, so no new dimension threshold or solution of Falconer is asserted. The criterion and stress tests are not claimed to be new in the literature without further comparison.

Sources

• S1 Krystal Taylor and Samantha Sandberg-Clark, Triangles in the plane and arithmetic progressions in thick compact subsets of R^d, Introduction. https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/triangles-in-the-plane-and-arithmetic-progressions-in-thick-compact-subsets-of-mathbb-rd/B5F9FC2F5A15CA0577BEDC96031B84CB. Catalogue formulation source.
• E1 Larry Guth, Alex Iosevich, Yumeng Ou and Hong Wang, On Falconer's distance set problem in the plane, Inventiones Mathematicae 219 $2020$, 779--830; arXiv:1808.09346, main theorem and the good/bad measure decomposition. https://arxiv.org/abs/1808.09346. Consulted 27 September 2026.
• E2 Xiumin Du, Yumeng Ou, Kevin Ren and Ruixiang Zhang, New improvement to Falconer distance set problem in higher dimensions, arXiv:2309.04103v2 (22 October 2024), Theorem 1.1 and Section 1. https://arxiv.org/abs/2309.04103. Consulted 27 September 2026; the version used includes dimension three.