On general densities and intersectivity

For a sequence of rectangles R=(Rk)k=1∞ in Nd and a subset FF of Nd, when the limit exists set d(F,R)=limk→∞|F∩Rk||Rk|. Suppose the subset EE of Nd has positive Banach density B(E)B(E). We give conditions on RR to ensure there exists a subset SS of Nd with d(S,R)≥B(E)d(S,R)≥B(E) such that for each finite subset {m1,…,mr}{m1,…,mr} of SS we have B(E∩(E+m1)∩⋯∩(E+mr))>0.B(E∩(E+m1)∩⋯∩(E+mr))>0.