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<p>For integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1&lt; k &lt; d-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>k</mi> <mo>&lt;</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(r \geqslant k+2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>⩾</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish new lower bounds on the maximum number of points in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\([n]^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> such that no <i>r</i> lie in a <i>k</i>-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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(d \geqslant 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.</p>

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On Subsets of Lattice Cubes Avoiding Affine and Spherical Degeneracies

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月16日

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  • Anubhab Ghosal,
  • Ritesh Goenka,
  • Peter Keevash

摘要

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.