… forming a dilate λ · A we will also make use of the operation of repeated addition λ ∗ A = {a1
+ ··· + aλ : a1,...,aλ ∈ A}. Moreover, we will need to be able to bound the size of sums of the …

For k prime and A a finite set of integers with| A|≥ 3 (k− 1) 2 (k− 1)! we prove that| A+ k·
A|≥(k+ 1)| A|−⌈ k (k+ 2)/4⌉ where k· A={ka: a∈ A}. We also describe the sets for which …

We consider the problem of sums of dilates in groups of prime order. It is well known that
sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of …

… the additive constant in Theorem 1.1 in terms of the additive constants coming from the
sums of the k − 1 dilates. We remind the reader that we are using induction on k, and that for …

… Z}$, the sum of dilates $A + \lambda \cdot A$ is defined by \[A + \lambda \cdot A = \{a + \lambda
a' : a, a' \in A\}.\] The basic problem on such sums of dilates asks for the minimum size of $|…

… We generalize this result for the sum of dilates A+r·B for two sets A … of dilates is one of the
important problems in additive combinatorics. Here we shall mainly consider the sum of dilates …

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = \{ \lambda a : a \in A\}$. Suppose $r, h\in
\mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A| \le …

… The first series of results in this paper concerns the size of the dilate set A+2·A. We present
a construction, the … In the next section we consider sums and dilates of a set with itself. …

Let $A$ be a subset of integers and let $2\cdot A+k\cdot A=\{2a_1+ka_2 : a_1,a_2\in A\}$. YO
Hamidoune and J. Ru\' e proved that if $k$ is an odd prime and $A$ a finite set of integers …

… In this paper, we consider sums of dilates of a finite set of integers. The next known lemma
shows that the size of a dilates sum remains invariant if we replace A by an affine transform of …