Kakeya-inspired rational spherical arrays: Achieving the lower bound for full-sphere coverage and capacity optimization

This paper introduces a Kakeya-inspired framework for optimizing spherical rational antenna arrays, establishing a Kakeya lower bound (KLB) that defines the theoretical minimum number of elements required to achieve complete spherical coverage. The Kakeya problem-unsolved for over a century until its recent resolution in three dimensions-provides the mathematical foundation for deriving these bounds, linking geometric measure theory to antenna deployment. Building on the recent developments in rational (non-integer) array theory, the proposed method formulates a closed-form relationship between the Kakeya coverage bound and the minimal directional pointings needed on a sphere, then realizes this bound through coprime-based rational element placement using Fibonaccidistributed surface sampling. Simulation results confirm that the proposed Kakeya-rational sphere achieves ≥ 90 % coverage with up to ≈ 75 % fewer elements than conventional integer-spaced arrays while improving sum-rate capacity. The framework also achieves faster global beam reconfiguration through a reduced time-to-full-coverage (TTFC), underscoring its potential for energy-efficient and high-capacity 6G and non-terrestrial communication systems. This work thus bridges a century-old mathematical milestone with practical antenna engineering, establishing the first implementation of Kakeya-derived lower bounds in spherical array design.