Additive bases of vector spaces over prime fields

It is shown that for any t > c p log n linear bases B 1 , …, B t of Z p n their union (with repetitions)∪ i = 1 t B i forms an additive basis of Z p n ; i.e., for any x e Z p n there exist A 1 ⊃ B 1 , …, A t ⊃ B t such that x = Σ i = 1 t Σ y e A i y .