On thin sets of primes expressible as sumsets

On Thin Sets of Primes Expressible as Sumsets

In this paper we will use the following notation. Given a set of positive integers S, we let S(x) denote the number of elements in S that are ≤ x, and we let |S| denote the total number of elements of S. Given two sets of positive integers A and B, we denote the sumset {a + b : a ∈ A, b ∈ B} by A + B; and so, the number of elements in A + B that are ≤ x will be (A + B)(x). For a finite set of i...

On sumsets of dissociated sets

in groups F2 is found. Using our approach, we easily prove a recent result of J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let Q be a set belonging to a sumset of two dissociated sets such that equation (1) has many solutions. We prove that in the case the large proportion of Q is highly stru...

Sumsets of sparse sets

Let σ be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c1x σ and at most c2x σ elements in the interval [1, x] for some constants c2 > c1 > 0 independent of x and each x ≥ x0. We prove that then the sumset A + A has more elements than A (counted up to x) by a factor c(σ) √ log x/ log log x for x large enough. An example showing that...

Sumsets of Sidon sets

Sieving Very Thin Sets of Primes, and Pratt Trees with Missing Primes

Suppose P is a set of primes, such that for every p ∈ P , every prime factor of p − 1 is also in P . We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x) for some c > 0, where c depends only on the smallest prime not in P . Our proof makes use of results connected with Artin’s primitive root conjecture.