Coordinate density of sets of vectors

Let n_o = 0,n₁ ≥1...,n₃ ≥1 be given natural numbers, J_i= ∑ t=0 0-1n_i+1,..., ∑ t=0 tn_t (i=J_i,...,s) and Π t=0 s Eⁿ_a=(x(^t),...., x(¹): n= ∑ i=1 s n₁ and if rε{lunate} J_i, then x(^u)ε{lunate}{0,..., q_i-J} A set R⊆∏_i³ = 1 E_ajⁱ² is said to be (m₁, ...,m₃)-dense (1≤m₁≤n) if there exist I₁⊆J₁ such that |L₁| = m₆ (i = 1,...,s) and |L^(I)(R)| = ∏_I⁵ = q_l^m1 where P^(I)(R) is the projection of R on the coordinate axes whose indices lie in I = ∪_i⁵ = _jL₁. In this paper we establish necessary and sufficient conditions for an arbitrary set R⊆∏⁵₁ = 1 E_a1ⁿ¹ with given |R| to be (m₁,...,m₃)-dense.