Non-degenerate spheres in three dimensions

Let P be a set of n points in ℝ³, and let k ≤ n be an integer. A sphere σ is k-rich with respect to P if |σ ∩ P| ≥ k, and is η-non-degenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂, σ contains more than η|σ ∩ P| points of P. We improve the previous bound given in [1] on the number of k-rich η-non-degenerate spheres in 3-space with respect to any set of n points in ℝ³, from O(n⁴/k⁵ + n³/k ³), which holds for all 0 < η < 1/2, to O*(n ⁴/k^11/2 + n²/k²), which holds for all 0 < η < 1 (in both bounds, the constants of proportionality depend on η). The new bound implies the improved upper bound O*(n ^58/27) ≈ O(n^2.1482) on the number of mutually similar triangles spanned by n points in ℝ³; the previous bound was O(n^13/6) ≈ O(n^2.1667) [1].