Problems

For a transcendental entire function, how fast can |f(z)| be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus M(r, f)?

The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate…

Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the…

For s ∈ (1/4,1) and any degree, the only W^s,1/s-minimizers among maps S^1 → S^1 are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4…

Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does ∑_n σ_n,ε n |hat f(n)|^2 → deg f hold for Holder maps below the threshold? No. For every 0 < α < 1/3 there is an f ∈…

For the three-dimensional paraboloid P_3 over a prime field in which -1 is not a square, the Fourier extension operator maps L^2 to L^r for r > 176/51 = 3.45098…, improving the exponent by combining a bilinear approach with point-line…

Vinzant conjectured, in a form later restated by Bandeira, that the 4M-4 threshold for injective complex phase retrieval is sharp. Part (1) holds: for A ∈ C^N × M with N = 4M-5 and i.i.d. standard complex Gaussian entries, the phase…

Near-optimal density thresholds forcing a measurable set in R^d to contain all sufficiently large similar copies of every n-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.

For a closed infinite set F ⊆ C, let μ(F) be the infimum of |z : |f(z)| < 1| over monic polynomials with zeros in F. Is μ(F) determined only by the transfinite diameter of F?