Discrete analysis, tilings, probability and entropy

Results in Discrete analysis, tilings, probability and entropy

• Selected result:

  The small-coefficient upper half of the Keller–Klein conjecture

  Partially resolvedKeller–Klein conjecture on Berry–Esseen bounds for Rademacher sums

  Mathematics · Probability and normal approximation · standalone Lean package

  For X = Σ vᵢεᵢ with independent uniform signs, Σ vᵢ² = 1 and |vᵢ| ≤ a ≤ 7/100, P(X ≤ x) − Φ(x) ≤ Φ(a) − 1/2 for every x ≥ 0: the upper half of the Keller–Klein conjecture in this small-coefficient range.

  Manuscript: The small-coefficient upper half of the Keller–Klein conjecture · Lean proof: The small-coefficient upper half of the Keller–Klein conjecture · Verification: The small-coefficient upper half of the Keller–Klein conjecture

• Selected result:

  The sharp level-one Fourier weight at density one eighth

  Determined exactlyLei Yu's problem on the level-1 Fourier weight W(1/8) of Boolean functions of density 1/8

  Mathematics · Discrete analysis · standalone Lean package

  Every Boolean function of mean 1/8 on {−1,1}^n has level-one Fourier weight at most 3/64, attained by a codimension-three subcube for n ≥ 3; equivalently, density-1/8 sets have average Hamming distance ≥ (n − 3)/2.

  Manuscript: The sharp level-one Fourier weight at density one eighth · Lean proof: The sharp level-one Fourier weight at density one eighth · Verification: The sharp level-one Fourier weight at density one eighth

• Sharp stability and exact cap optima for Rademacher sums

  Determined exactlyExact Kolmogorov-distance cap optimum and support-gap stability for Rademacher sums

  Mathematics · Probability and normal approximation · standalone Lean package

  A large support gap forces an odd flat core with the sharp distance bound 2 − 2√(1 − (a − x)²); support-gap near-optimizers are stable with optimal constant √(2+√2) at every cap; and, conditionally on the completed Keller–Klein comparison, the maximal Kolmogorov distance is exact for 1/√5 ≤ a ≤ 1/√3.

  Manuscript: Sharp stability and exact cap optima for Rademacher sums · Lean proof: Sharp stability and exact cap optima for Rademacher sums · Verification: Sharp stability and exact cap optima for Rademacher sums

• Structural selfadhesivity does not imply a numerically selfadhesive realization

  Answered negativelyBoege's selfadhesivity question: structural vs numerical selfadhesive polymatroids

  Mathematics · Entropy and conditional independence · standalone Lean package

  An explicit five-variable conditional-independence model with 26 of the 80 elementary CI statements is structurally selfadhesive at every overlap, yet no real polymatroid inducing it is selfadhesive in Matúš's numerical sense.

  Manuscript: Structural selfadhesivity does not imply a numerically selfadhesive realization · Lean proof: Structural selfadhesivity does not imply a numerically selfadhesive realization · Verification: Structural selfadhesivity does not imply a numerically selfadhesive realization

• A quartic entropy deficit for random rankings

  SharpenedLeighton–Moitra entropy conjecture for rankings: an explicit quartic ε⁴ entropy deficit

  Mathematics · Probability and combinatorics · standalone Lean package

  Every probability distribution on rankings of n labels in which each pairwise comparison has probability at distance at least ε > 0 from 1/2 has entropy at most (1 − ε⁴/144) log(n!), for every n.

  Manuscript: A quartic entropy deficit for random rankings · Lean proof: A quartic entropy deficit for random rankings · Verification: A quartic entropy deficit for random rankings

• Sharp universal Rényi threshold for iid sums on finite Sidon sets

  Determined exactlyRényi entropy of iid sums on Sidon sets: for which orders is uniform input optimal?

  Mathematics · Additive combinatorics and entropy · standalone Lean package

  For Rényi order α > 1, the uniform law maximizes H_α(X+X') for iid X, X' on every nonempty finite strong Sidon (B₂) set exactly when 2^(α−1) ≤ 2α − 1, i.e. α ≤ α_c ≈ 3.65986.

  Manuscript: Sharp universal Rényi threshold for iid sums on finite Sidon sets · Lean proof: Sharp universal Rényi threshold for iid sums on finite Sidon sets · Verification: Sharp universal Rényi threshold for iid sums on finite Sidon sets

• A 64-point spectral tile in F₂⁹ with no periodic complement

  CounterexampleFerguson–Iosevich Question 8.1: subspace tiling complements of spectral sets in F₂⁹

  Mathematics · Spectral sets and tilings · standalone Lean package

  An explicit 64-point set E ⊂ F₂⁹ is spectral and tiles F₂⁹, but its tiling complements are exactly the 512 translates of one aperiodic 8-point set, so no subspace complements E and F₂⁹ fails the periodic tiling property.

  Manuscript: A 64-point spectral tile in F₂⁹ with no periodic complement · Lean proof: A 64-point spectral tile in F₂⁹ with no periodic complement · Verification: A 64-point spectral tile in F₂⁹ with no periodic complement