For arbitrary c0>0, 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 Tk(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.
