Monochromatic sumsets

Monochromatic sumsets

LEMMA . At most (kn)'g"ulk k-sets Sc [ n ] have IP(S)I <u . Proof. Let a,< . . . < a k denote the elements of S. Call i doubling if P(a, , . . ., a i) has double the size of P(a, , . . ., a i 1 ) . There are at most lg u doubling i. Hence there are at most k'g" choices for doubling positions i and at most n'g" choices for the values a i . If i is not doubling then a i = x y, where x, y E P(a, ,...

On Various Restricted Sumsets

For finite subsets A1, . . . , An of a field, their sumset is given by {a1+ · · ·+an : a1 ∈ A1, . . . , an ∈ An}. In this paper we study various restricted sumsets of A1, . . . , An with restrictions of the following forms: ai − aj 6∈ Sij , or αiai 6= αjaj , or ai + bi 6≡ aj + bj (mod mij). Furthermore, we gain an insight into relations among recent results on this area obtained in quite differ...

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...

Additive Structures in Sumsets

Suppose that A and A are subsets of Z/NZ. We write A + A for the set {a + a : a ∈ A and a ∈ A′} and call it the sumset of A and A. In this paper we address the following question. Suppose that A1, ...,Am are subsets of Z/NZ. Does A1 + ...+ Am contain a long arithmetic progression? The situation for m = 2 is rather different from that for m ≥ 3. In the former case we provide a new proof of a res...

Sumsets of Sidon sets