Kowalski (1991) obtained the quantity c(k, r, a), which satisfies the condition max {m(A∩f−i1 A∩ ⋯ ∩ f−ir A): 1 ⩽ i1 < ⋯ < ir ⩽ k} ⩾ c(k, r, a) for any dynamical system (X,, m, f) and every set A∈, with m(A) = a.
Our present purpose is to prove the following theorem: for every ergodic aperiodic endomorphism and for every integer k, and for each real a, 0 < a < 1, such that (k + 1)a is not an integer, there exists a set A with m(A) = a fulfilling the equality m(A∩f−i1 A∩⋯∩f−ir A) = c(k, r, a), for any sequence 1 ⩽ i1 < ⋯ < ir ⩽ k, r = 1, …, k.