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 $A\subset \mathbb{F}_q$ we have $$|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.$$ This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that $$\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.$$