On Subsets of Lattice Cubes Avoiding Affine and Spherical Degeneracies
<jats:p> For integers <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$1< k < d-1$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>k</mml:mi> <mml:mo><</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$r \geqslant k+2$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>⩾</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , we establish new lower bounds on the maximum number of points in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$[n]^d$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> such that no <jats:italic>r</jats:italic> lie in a <jats:italic>k</jats:italic> -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 <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$d \geqslant 4$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems. </jats:p>