Home > Combinatorics > IMC 2012 Problem 10

IMC 2012 Problem 10


Let {c \geq 1} be a real number. Let {G} be an abelian group and let {A \subset G} be a finite set satisfying {|A+A|\leq c|A|} where {X+Y=\{x+y : x \in X , \ y \in Y\}} and {|Z|} denotes the cardinality of {Z}. Prove that

\displaystyle |\underbrace{A+A+...+A}_{k \text{ times}}| \leq c^k |A|

for every positive integer {k}.

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