Sum-product estimates over arbitrary finite fields

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $Asubset mathbb{F}_q$ we have [|(A-A)^2+(A-A)^2|gg |A|^{1+frac{1}{21}}.] This can be viewed as the ErdH{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that [max{|A+A|, |A^2+A^2|}gg |A|^{1+frac{1}{42}}, ~|A+A^2|gg |A|^{1+frac{1}{84}}.]