A lower bound for the size of a sum of dilates (Q2869046)

Let \(A\) be a finite non-empty set of integers. The \(k\)-dilate of the set \(A\) is the set of all integers of the form \(ka\), where \(a\in A\). An asymptotically sharp lower bound on the size of sumsets of the form \(u_1A+u_2A+\dots+u_hA\) was obtained by \textit{B. Bukh} [Comb. Probab. Comput. 17, No. 5, 627--639 (2008; Zbl 1191.11007)]. In the case of binary linear forms \(uA+vA\), where \(u,v\) are nonzero integers it is enough to consider normalized linear forms satisfying \(v\geq |u|\geq 1\) and \((u,v)=1\) (cf. \textit{M. B. Nathanson} et al. [Acta Arith. 129, No. 4, 341--361 (2007; Zbl 1156.11011)]). \textit{J. Cilleruelo} et al. [Comb. Probab. Comput. 18, No. 6, 871--880 (2009; Zbl 1200.11007)] proved that if \(u=1\) and \(v\) is a prime and \(A\) is a finite set of integers with \(|A|\geq 3(v-1)^2(v-1)!\) then \(|A+v \cdot A|\geq (v+1)|A|-\lceil v(v+2)/4\rceil\) thereby confirming a conjecture of Cilleruelo, Silva and Vinuesa [\textit{J. Cilleruelo} et al., J. Comb. Number Theory 2, No. 1, 79--89 (2010; Zbl 1245.11029)] formulated for a general \(v\). The case \(u=1\) and \(v\) is a power of a prime or \(v\) is a product of two primes was settled by \textit{S.-S. Du} et al. [Electron. J. Comb. 21, No. 1, Research Paper P1.13, 25 p. (2014; Zbl 1308.11010)]. \textit{Y. O. Hamidoune} and \textit{J. Rué} [Comb. Probab. Comput. 20, No. 2, 249--256 (2011; Zbl 1231.11013)] proved that if \(v\) is an odd prime and \(|A| > 8v^v\) then \(|2 \cdot A + v \cdot A|\geq (v+2)|A|-v^2-v+2\) provided \(v\) is an odd prime. In the present paper this result is extended to the case when \(v\) is a power of an odd prime or a product of two distinct odd primes.