Bourgain’s periodic Strichartz conjecture

Establish that for the periodic (torus) Schrödinger equation in dimension d, the optimal Strichartz constant at the critical exponent satisfies C(2(d+1)/(d−1), N) ≤ C_ε N^ε uniformly in N.

Background

This conjecture asserts sharp ℓ^2-based decoupling-type control of periodic Schrödinger solutions at the critical L^q exponent, matching random-matrix bounds and predicting universal N^{o(1)} growth.

It was proved for d=2,3 via even-integer power methods but remained open for higher d until Bourgain–Demeter’s decoupling, which now yields the stated bound (up to N^ε) in all dimensions; the conjecture records its original formulation.