On dilates sums

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Let A be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained by Bukh. The unique known exact bound concerns the sum |A+k· A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let k be an odd prime and assume that |A|>8kᵏ. A corollary to our main result states that |2· A+k· A|≥ (k+2)|A|-k²-k+2. Notice that |2· P+k· P|=(k+2)|P|-2k, if P is an arithmetic progression.