Bourgain–Kakeya conjecture for fat arithmetic progressions

Establish that for T = N^α with 1 < α < 2, after partitioning {N+1,…,2N} into intervals I of length ≍ N T^{−1/2} and forming their associated fat arithmetic progressions I*, any choice of translates t_I ∈ [0,T] satisfies ∫_0^T (∑_I 1_{I* + t_I})^{p/2} ≤ C(ε,α) N^ε ∫_0^T (∑_I 1_{I*})^{p/2} for all p ≥ 1 and ε > 0.

Background

This number-theoretic analogue of Kakeya controls overlaps among large-value regions generated by short Dirichlet polynomials (approximate geometric series). Bourgain showed Montgomery’s ℓ^q conjecture implies this inequality and, in turn, that it implies the classical Kakeya conjecture.

Proving this would preclude pathological overlap patterns for the fat progressions and sharpen the understanding of Dirichlet large-value behavior tied to wave-packet–like structures.