Falconer Distance Conjecture, Bochner–Riesz Conjecture and 8 More

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |ϕ'|^s is area-integrable for 4/3 < s < 4. The sharp universal integral-means identity is B_𝒮(t)=|t|-1 for t ≤ −2. A strict bound B_b(-1) < 1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter. (Lean)

Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E ⊂ ℝ^d with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure. (Lean)

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in L log L(𝕋) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the L log L scale. (Lean)

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on L^p(ℝ³) for every 1 ≤ p ≤ ∞ whenever δ > max {3|1/p-1/2|-1/2,0}.

Constructs polynomials with N consecutive coefficients in {-1,1} whose modulus is (1+o(1))√N uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture. (Lean)

Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface Σ ⊂ ℝ³, including surfaces with boundary, the extension operator is bounded from L^p(Σ) to L^p(ℝ³) for every p > 3.

Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2 < p < ∞, with Sobolev regularity above max {0,1-3/p}. In particular, the critical L3 estimate holds with every positive Sobolev loss.

Proves almost-everywhere convergence e^itΔf → f as t↓0 for every f ∈ H^n/(2(n+1))(ℝⁿ) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H1/3.

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2 ≤ n ≤ d-2, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds. (Lean)

Proves maximal and annular r-variation bounds, for every r > 2, from complex L³(ℝ²)× L³(ℝ²) to L^3/2(ℝ²). The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric L³× L³× L³ point. (Lean)