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It is established that there exists an absolute constant    c  >  0    such that for any finite set   A   of positive real numbers|  A  A  +  A  |   ≫    |  A  |3  2   +  c    .    On the other hand, we give an explicit construction of a finite set    A  ⊂  R    such that    |  A  A  +  A  |  =  o  (  |  A   |  2   )   , disproving a conjecture of Balog.

On the size of the set A A + A