On subsets of lattice cubes avoiding affine and spherical degeneracies

For integers $1 < k < d-1$ and $r \ge k+2$, we establish new lower bounds on the maximum number of points in $[n]^d$ 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 Erd\H{o}s and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $d \geq 4$. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.