Abstract
Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then \[ |A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil \] provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We also establish the inequality $|A+4\cdot A|\geq5|A|-6 $ for $|A|\geq5$.
Highlights
For finite subsets A1, . . . , Ak of Z, their sumset is given by A1 + · · · + Ak = {a1 + · · · + ak : a1 ∈ A1, . . . , ak ∈ Ak}, which is denoted by kA if A1 = · · · = Ak = A
Lower bounds for |k1 · A + k2 · A + · · · + kl · A| were investigated by various authors [1, 2, 8, 9]
We have completed the proof of Theorem 3
Summary
For l = 2 there are better quantitative results in this direction, see [3, 4, 5, 7, 9] It was conjectured in [4] that for any k ∈ Z+ if |A| is sufficiently large . Motivated by the preprint form of our paper posted to arXiv, Ljujic [6] obtained similar results for |2 · A + k · A| with k a prime power or a product of two distinct primes. Let p1 and p2 be distinct primes and k = p1p2. By Theorem 1, if k = 4 (1) holds when |A| 216. We remark that the lower bound given in (1) is optimal when |A| is large enough. Our key new idea is to employ Chowla’s theorem to handle the case when k is a prime power, and use a lemma similar to Chowla’s theorem to handle the case when k is a product of two distinct primes
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