Non-Degenerate Spheres in Three Dimensions

Let P be a set of n points in R, and k ≤ n an integer. A sphere σ is k-rich with respect to P if |σ ∩ P | ≥ k, and is η-nondegenerate, 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-nondegenerate spheres in 3space with respect to any set of n points inR, from O(n/k+n/k), which holds for all 0 < η < 1/2, to1 O∗(n4/k11/2+n2/k2), which holds for all 0 < η < 1 (in both bounds, the contants of proportionality depend on η). The new bound implies the improved upper bound O∗(n58/27) ≈ O(n) on the number of mutually similar triangles spanned by n points in R; the previous bound was O(n) ≈ O(n) [1].