A new type of Littlewood-Paley partition

We define a partition of Z into intervals {I _j} and prove the Littlewood-Paley inequality ‖f‖ _p ≦C _p‖Sf‖ _p , 2≦p<∞. Heref is a function on [o, 2π) and (Sf = (sum |Delta _j |^2 )^{1/2} , hat Delta j = hat fchi _{Ij} ) . This is a new example of a partition having the Littlewood-Paley property since the {I _j} are not of the type obtained by iterating lacunary partitions finitely many times.