For arbitrary c00, if A is a subset of the primes less than x with cardinality δx（logx）-1, δ≥（logx）-c0, then there exists a positive constant c such that the cardinality of A＋A is larger than cδx（loglogx）-1.
In this paper, we establish a quite general mean value result of arithmetic functions over short intervals with the Selberg-Delange method and give some applications. In particular, we generalize Selberg's result on the distribution of integers with a given number of prime factors and Deshouillers-Dress-Tenenbaum's arcsin law on divisors to the short interval case.
Let Fp be the finite field of p elements with p prime.If A is a subset of Fp and g is an element of F＊p with order ν,then max{｜A ＋ g·A｜,｜A·A｜} （ν/（ν ＋ ｜A｜2） ）1/12｜A｜13/12.
