Some remarks on the Balog-Wooley decomposition theorem and quantities   D^+, D^\times
  In the paper we study two characteristics D^+ (A), D^\times (A) of a set A which play important role in recent results concerning sum-product phenomenon. Also we obtain several variants and improvements of the Balog-Wooley decomposition theorem. In particular, we prove that any finite subset of real numbers can be split into two sets with small quantities D^+ and D^\times. 
Ilya D. Shkredov
math.CO math.NT