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{Omega(1)}) lower bound for an explicit range counting problem of n{3} convex polygons in mathbb{R}{2} (each with n{tilde{O}(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-delta}). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.