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October 12, 2025Open Access

On subsets of lattice cubes avoiding affine and spherical degeneracies

Key Points

New lower bounds are established for the maximum number of points avoiding $k$-dimensional affine subspaces in $[n]^d$.
The randomised construction improves Thiele's bound for the no-four-on-a-circle problem, offering a novel approach to counting configurations.
The deletion method is utilized, integrating results from number theory and incidence geometry to enhance existing bounds in higher dimensions.
Improvements are made on earlier results by Sudakov-Tomon and Lefmann, indicating significant advancements in the area of combinatorial geometry.

Abstract

For integers 1 < k < d-1 and r k+2, we establish new lower bounds on the maximum number of points in nᵈ 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 4. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.

On subsets of lattice cubes avoiding affine and spherical degeneracies

Key Points

Abstract

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