A bound on the multiplicative energy of a sum set and extremal   sum-product problems
  In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool is a new result which provides a nontrivial upper bound on the multiplicative energy of a sum set or difference set. 
Oliver Roche-Newton and Dmitry Zhelezov
math.CO math.NT