On certain properties of sets with Baire property (Q1611390)

Main result: Let \(I\) be a non-empty at most countable set. Let \((a,A)\) and \((a_i,A_i)\) \((i\in I)\) be point-set pairs in \(\mathbb{R}^r\), such that \(A\) is of second category at \(a\) and for each \(i\) in \(I\), \(A_i\) is residual at \(a_i\), so \(B(a_i;\delta_i)\setminus A_i\) is of first category for some \(\delta_i> 0\). Also, for each \(i\) in \(I\), let \((d^{(i)}_n)\) be a sequence in \(\mathbb{R}^r\) converging to \(a_i- a\). Then there is a first category set \(D\), and for each \(i\) in \(I\) given \(0< \eta_i< \delta_i\) there is a positive integer \(k_i\) such that \(x\in B(a;\eta_i)\setminus D\) implies \(x+ d^{(i)}_n\in A_i\cap B(a_i;\delta_i)\) for every \(n\geq k_i\).