Polynomial Data Structure Lower Bounds in the Group Model

Proving super-logarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early 80's. We prove a polynomial (n
^Ω(1)
) lower bound for an explicit range counting problem of n
³
convex polygons in \mathbbR
²
(each with n
^Õ̃(1)
facets/semialgebraic-complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine approximation, pseudorandomness, and compressed sensing-in particular, on the existence and partial derandomization of optimal binary compressed sensing matrices in the polynomial sparsity regime (k=n
^1-δ
). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.