Multiple returns under a bounded number of iterations (Q1314410)

The author introduces preliminarily the ingredient \(c(k,r,a)\) as follows: For any dynamical system \((X,{\mathcal B},m,f)\) and for every set \(A \in {\mathcal B}\) with \(m(A) =a\), there exists \(c(k,r,a)\) such that \[ \max \{m(A \cap f^{-i_ 1}A \cap \cdots \cap f^{-i_ r} A):1 \leq i_ 1 \leq \cdots \leq i_ r \leq k \} \geq c(k,r,a). \] The main result of the paper consists of the theorem: For every ergodic aperiodic endomorphism and positive integer \(k\), and for each \(a\), \(0<a<1\), such that \((k+1)a\) is not an integer, there exists a set \(A\) with \(m(A)=a\) and satisfying the equation \[ m(A \cap f^{-i_ 1}A \cap \cdots \cap f^{-i_ r} A)=c(k,r,a) \] for any sequence \(1 \leq i_ 1 \leq \cdots \leq i_ r \leq k\), \(r=1,\dots,k\).