On Subsets of Lattice Cubes Avoiding Affine and Spherical Degeneracies

Abstract For integers $$1< k < d-1$$ 1 < k < d - 1 and $$r \geqslant k+2$$ r ⩾ k + 2 , we establish new lower bounds on the maximum number of points in $$[n]^d$$ [ n ] d such that no r lie in a k -dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erdős and Purdy, improving Thiele’s bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $$d \geqslant 4$$ d ⩾ 4 . In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.