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Let      G G       be the group of integers of a      p p      -series field and suppose that      S S       is a character series on      G G      . If      N 1 {N_1}      ,      N 2 , …  {N_2}, \ldots       is any sequence of integers and if      S p N j →  0 {S_{{p^{{N_j \to 0       a.e. on      G G      , as      j →  ∞  j \to \infty      , then      S S       will be the zero series provided      S S       never diverges unboundedly.

Uniqueness and quasimeasures on the group of integers of a 𝑝-series field