ON THE LOWER BOUND FOR THE CARDINALITY OF SUMSETA + 5 · A

Let A and B be two non-empty finite subsets of Z. We consider the dilated sumset A+k·B = {a+kb : a ∈ A, b ∈ B}, where k ∈ Z+. In 2010, Cilleruelo et al. proved that |A+3 ·A| ≥ 4|A|−|A|, where A is finite subset of Z. They proposed the following conjecture. If A is a non-empty finite subset of Z with arbitrarily large cardinality and k ∈ Z+, then |A+k·A| ≥ (k+1)|A|−⌈(k2+2k)/4⌉. In 2009, Cilleruelo et al. proved this conjecture for k = p, with |A| ≥ 3(k − 1)2(k − 1)!. In this article, we prove that this conjecture is true for smaller as well as larger value of |A| with |A| ≥ 6 and k = 5.