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.
Let fk（n） be the characteristic function of n with Ω（n） = k, and T k（x,α）=∑n≤xfk（n）e（nα）.The main purpose of this paper is to establish a Bombieri-type mean-value theorem for Tk（x, α）, via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions. The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method
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.
Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
