New results on sum-products in R

We improve a previous sum–products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure energy sum–products results, improving a result of Balog and Wooley, in particular.