Distinct Distances on Non-Ruled Surfaces and Between Circles

We improve the current best bound for distinct distances on non-ruled algebraic surfaces in $${\mathbb {R}}^3$$ . In particular, we show that n points such a surface span $$\Omega (n^{32/39-{\varepsilon }})$$ distances, any $${\varepsilon }>0$$ Our proof adapts of Székely planar case, which is based crossing lemma. As part our surfaces, also obtain new results between circles Consider two point sets respective sizes m and n, each set lies circle characterize cases when number can be $$O(m+n)$$ This includes configuration with small distances. other prove (\min {\{m^{2/3}n^{2/3},m^2,n^2\}})$$