Difference-set sum conjecture for Dirichlet polynomials

Prove that for a Dirichlet polynomial D(t) = ∑_{n=N}^{2N} b_n e^{it log n} with |b_n| ≤ 1 and any 1-separated set 𝒯 ⊂ [0,T], one has ∑_{t1,t2∈𝒯} |D(t1 − t2)|^2 ≤ T^{o(1)} (|𝒯| N^2 + |𝒯|^2 N).

Background

Heath-Brown’s approach uses a majorant principle on difference sets to control pairwise differences of times; assuming the majorant Dirichlet sum bound, this conjecture would be sharp and would imply strong large-value bounds when the time set exhibits additive structure.

The conjecture is proven in some ranges (e.g., N ≤ T ≤ N^{3/2}) via induction on N and careful analysis of the majorant; the general case remains open.