Perplexity
Perplexity develops AI tools for search, browsing, research, content creation, organization, and workflow automation, including perplexity.ai, Comet, and Computer.
Binary Edit Distance Requires L1 Distortion of exp(Θ(√(log d · log log d)))
An OpenAI preprint proves that, for every sufficiently large length cap d, there is a finite family of binary words of length at most d whose pairwise edit distances require distortion of at least exp(c√(log d · log log d)) in any embedding into L1, for a positive absolute constant c. The lower bound applies to a constructed family, not to each pair of words or to the computational cost of measuring their edit distance.
Uniform Riesz Transform Bounds Force Rectifiable Pieces in Higher Codimension
An OpenAI preprint argues that, in a specified range of dimensions, uniform L² bounds on all hard truncations of a measure’s Riesz transform force uniform rectifiability, provided the measure is Ahlfors-David regular. In geometric terms, every admissible ball must contain a substantial portion of the measure on a Lipschitz image of an n-dimensional ball. The proof links the analytic bound to controlled failures of flatness across scales, rather than requiring the support to be flat at every location.
The Triangular Lattice Minimizes Repulsive Energy Beyond Periodic Configurations
An OpenAI preprint argues that the unit-density triangular lattice minimizes energy not only among lattices but among all locally finite point configurations of density one, for a defined class of repulsive interactions. Its proof constructs an exact auxiliary-function certificate: a global lower bound for competing configurations that matches the lattice’s energy. The manuscript says the result extends from Gaussian interactions to completely monotone functions, while stressing that finite arithmetic checks support—but do not replace—the proof’s infinite-tail and continuous sign arguments.
The Triangular Lattice Minimizes Planar Coulomb Energy Beyond Periodic Patterns
An OpenAI preprint on planar Coulomb renormalized energy claims that the triangular lattice attains the minimum not only among periodic arrangements but across the full admissible class, including non-periodic fields. The manuscript establishes the lower bound by first treating a finite system on a torus, then transferring it to the plane; its proof combines a global comparison potential, Voronoi-sector geometry and computer-assisted inequalities. The claim is a bound on the energy, not a classification of every minimizing arrangement.
A Missing Factor Reverses the Claimed Simplex Counterexample
An OpenAI preprint argues that the product of two 10-dimensional simplices exceeds the normalized projection-body volume of a 20-dimensional simplex, presenting the result as a counterexample to the simplex maximum. But the manuscript’s stated benchmark and comparison are inconsistent: correcting the benchmark introduces a missing factor of 20 and makes the product’s score lower, not higher. The calculation therefore does not establish the claimed counterexample or the larger-dimensional extension built on it.
Every Sufficiently Large Degree Admits an Ultraflat Real Littlewood Polynomial
An OpenAI manuscript dated October 5, 2026, claims that for every sufficiently large length \(N\), one can choose \(+1\) and \(-1\) coefficients so that the resulting real Littlewood polynomial has modulus uniformly as close to \(\sqrt N\) as desired at every point on the unit circle. The paper’s argument first builds a nearly flat continuous wave, then uses its energy to show the wave’s Fourier coefficients are close on average to signs and a discrepancy lemma to round them with uniformly controlled error.
Centered Disk Maximal Functions Admit a Global Endpoint Gradient Bound
A September 2026 manuscript claims an endpoint gradient bound for the centered disk maximal operator in the plane: the integral of the maximal function’s gradient is at most an absolute constant times the integral of the input’s gradient. The result applies to real-valued functions with integrable values and weak first derivatives, without requiring radial symmetry or compact support. The authors say the proof overcomes the endpoint failure of the usual maximal-function estimate by combining cancellation with scale-by-scale bookkeeping to control changes in the radius that produces the largest average.
Koebe’s Circle-Domain Conjecture Holds Without Boundary or Countability Assumptions
An OpenAI manuscript claims to prove Koebe’s circle-domain conjecture: every non-empty connected open region on the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points, without assumptions on boundary smoothness or the number of components. Its central challenge is showing that finite approximations by round holes do not acquire non-round defects in the limit. The proof combines energy estimates, path-based controls and a contour argument to rule out those defects; the manuscript establishes existence, not uniqueness or convergence of arbitrary finite approximations.
A Scale-Independent L³ Bound for the Dyadic Triangular Hilbert Form
An OpenAI preprint proves a scale-independent \(L^3\) bound for a dyadic triangular Hilbert form built from three functions sharing variables around a cycle. The authors show that the sum of the absolute local contributions is at most 40 times the product of the functions’ \(L^3\) norms, regardless of which finite set of scales is used. Their proof charges each contribution to a nonnegative increase in matrix energy, then telescopes those increases across scales.
The Triangular Hilbert Transform Is Bounded Uniformly Across All Cutoffs
The preprint *The Maximal Triangular Hilbert Transform at the Symmetric Point* claims a uniform \(L^3\times L^3\to L^{3/2}\) bound for the triangular Hilbert transform, with the supremum taken over all inner and outer cutoffs at once. Its proof handles cutoff choices that vary across output points by controlling the cost of each entry’s switches, rather than paying for the total number of scales. The manuscript also claims almost-everywhere and \(L^{3/2}\) convergence as both cutoffs move outward.
A New Falconer Theorem Claims Positive-Measure Distances Above d/2
An OpenAI manuscript dated September 23, 2026, claims that any compact set in ℝᵈ with Hausdorff dimension greater than d/2 determines a set of distances with positive Lebesgue measure. The result addresses the unpinned Falconer distance conjecture in every integer dimension d ≥ 2, but does not cover the equality case or distances measured from a fixed point. The proof combines frequency estimates with a multiscale argument to obtain a distance measure with an integrable density.
Ellipsoids Minimize Projection-Body Volume Ratios in Dimensions Four and Higher
An OpenAI preprint claims a sharp solution to Petty’s projection-volume problem in every dimension \(n\ge4\): among convex bodies with fixed volume, ellipsoids—and only ellipsoids—minimize the volume of the body defined by their shadows. The manuscript reduces the global inequality to a bound on pairings of measures, with the hardest step controlling directional variation through higher spherical-harmonic components. Its geometric examples illustrate the result but do not establish the proof.
A Fixed Distortion Gap Cuts Lp Embedding Dimension Below Every Polynomial
An OpenAI preprint argues that any finite set of \(n\) points in a real \(L_p\) space, for fixed \(1<p<\infty\) with \(p\ne2\), can be embedded in finite-dimensional \(\ell_p\) using fewer than \(n^\varepsilon\) coordinates for every fixed \(\varepsilon>0\), provided any fixed distortion greater than 1 is allowed. The bound is not uniform as distortion approaches 1: the paper contrasts it with a quadratic coordinate requirement for exact preservation.
Logarithmic Brunn–Minkowski Holds for All Origin-Symmetric Convex Bodies
An OpenAI preprint proves the logarithmic Brunn–Minkowski conjecture for origin-symmetric convex bodies in every positive dimension. It shows that when support bounds are combined by a geometric average and used to define a new body, that body’s volume is at least the geometric mean of the original volumes. The proof approximates volume with smoothed integrals and establishes their log-concavity through a variance bound.
The Symmetric Mahler Bound Has Equality Exactly for Hanner Bodies
An OpenAI preprint on the symmetric Mahler problem claims that, in dimension \(n\), the product of the volumes of an origin-symmetric convex body and its polar is at least \(4^n/n!\). It says equality holds exactly for linear images of Hanner polytopes, a family built recursively from intervals using Cartesian products and convex-hull joins. The paper’s proof turns feasibility probabilities into a volume bound, then uses a metric median property to identify the equality cases.
Fourier Extension Is Bounded for Every Exponent Above Three
An OpenAI manuscript proves a diagonal Fourier extension estimate for every finite \(p>3\) on compact, smooth surfaces in three dimensions with positive curvature: the wave’s \(L^p\) norm across space is bounded by the input’s \(L^p\) norm on the surface. The authors build the global bound by combining packet-propagation and Kakeya estimates, using a local density inequality to connect them. The endpoint \(p=3\) is not covered.
A Sharp Three-Dimensional Kakeya Bound Tracks Sparse Tube Density
An OpenAI manuscript claims a sharp three-dimensional Kakeya maximal estimate, allowing a δ⁻ε loss as tubes become thinner. Its central claim is density-sensitive: the bound must hold even when only a sparse subset of each tube is retained, with the resulting union estimate weakening cubically in the retained fraction. The manuscript’s proof argues that near-extremal tube families must recur in a constrained geometric pattern across scales, then uses projection and information estimates to rule out the configurations that would violate the bound.
Three-Dimensional Wave Estimate Allows Arbitrarily Small Positive Loss
An OpenAI preprint claims a spacetime \(L^3\) smoothing estimate for the three-dimensional wave equation, but its derivation depends on a difficult estimate for a single frequency band that the later frequency summation does not prove. Conditional on that bound, any positive loss of derivatives makes the sum converge; the argument does not establish the zero-loss endpoint. The same qualification applies to the derived range for finite \(p>2\), whose regularity boundary is also excluded.
Schrödinger Waves Converge Pointwise at the Sobolev Endpoint in Dimensions Three and Higher
The paper claims pointwise convergence for the free Schrödinger equation at the exact Sobolev regularity threshold in every fixed dimension \(n \ge 3\). For data in \(H^{n/(2(n+1))}(\mathbb{R}^n)\), it first removes Gaussian damping and then takes time to zero, recovering the datum’s Lebesgue value on one full-measure spatial set that works for every time in the stated interval. The proof’s endpoint challenge is to sum frequency estimates without losing regularity.
Univalent Maps Have a Uniform Reciprocal-Derivative Growth Gap Below One Quarter
An OpenAI preprint proves that the average reciprocal derivative of a normalized univalent map grows more slowly near the unit-circle boundary than Kraetzer’s proposed one-quarter exponent permits. The bound holds uniformly across all such maps, with an exponent at least some fixed positive amount below \(1/4\); the paper establishes that a gap exists but does not calculate its size.
Bounded Treewidth Guarantees Finite-Distortion L1 Embeddings
An OpenAI preprint proves that every finite connected graph with bounded treewidth embeds into finite-dimensional \(L_1\) with distortion bounded by a constant determined by the tree-decomposition bag size—not by the graph’s number of vertices or the length of its decomposition. The paper’s central challenge is ensuring that distant vertex pairs remain separated, not merely that the embedding does not stretch distances too far. It addresses this with weighted cuts, using reservations and anchor cuts to preserve separation across the graph.
Short-Segment Hilbert Transforms Remain Bounded Under Lipschitz Direction Changes
An OpenAI preprint proves a uniform \(L^2\) bound for a Hilbert transform whose direction varies across the plane, under a short-segment restriction. The authors show that when the direction field is Lipschitz, the bound holds uniformly across truncations, with constants independent of the field and the input; the direction is fixed at each output point along the segment. Their proof uses a frequency-localized gain to control contributions as it reassembles the estimate across directions.
Planar Schrödinger Evolution Converges at the H¹ᐟ³ Endpoint
A September 2026 OpenAI manuscript claims that planar Schrödinger evolution converges almost everywhere to its initial data for functions with exactly \(1/3\) of a Sobolev derivative, reaching a boundary where earlier results required more regularity. The authors’ proof rests on a local \(L^1\) maximal estimate that controls the evolution uniformly over time at that endpoint. They define the evolution through a Gaussian-regularized Fourier integral and show the convergence holds on one full-measure set for all times approaching zero.
Near-Full-Measure Sets Can Avoid Every Copy of a Fixed Geometric Progression
An October 5, 2026 OpenAI manuscript claims that, for any fixed ratio between zero and one, a compact subset of the unit interval can have measure arbitrarily close to one while containing no translated, nonzero-scaled or reflected copy of the corresponding infinite geometric progression. The result concerns the entire sequence: positive measure guarantees copies of every finite prefix, but the manuscript constructs a set in which every such infinite copy has at least one term missing. Its proof organizes small holes to catch every center and scale, then uses convergence of the progression to cover centers missed by finite tests.