Lemma 10 ([Vector-valued hashing]). Let H be a finite-dimensional vector space over , write χ_q(u)=(-1)^ q(u) for q∈H^* and u∈H , let V be a real Hilbert space, let s≥1 be an integer, and let c(q,z)∈ V, q∈H^*, z∈^s. Set w_q=∑_ z∈^s …

Lemma 10 ([Vector-valued hashing]). Let H be a finite-dimensional vector space over , write χ_q(u)=(-1)^ q(u) for q∈H^* and u∈H , let V be a real Hilbert space, let s≥1 be an integer, and let c(q,z)∈ V, q∈H^*, z∈^s. Set w_q=∑_ z∈^s c(q,z)^2, M=∑_ q∈H^* w_q. Choose L,J_1,…,J_s independently and uniformly in H , and define J(q)=(q(J_1),…,q(J_s)), U(z)=∑_ q∈H^* χ_q(L)c(q,z+ J(q)). Then ∑_ z∈^s U(z)^2=M. For every ζ>0 , there are γ>0 and s_0∈ℕ , depending only on ζ , such that for every s≥s_0 , _ q∈H^* w_q≤γM _ z∈^s U(z)^2≤ζM.