Anubhab Ghosal

Open access Jul 2026

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>

Anubhab Ghosal, Ritesh Goenka, Peter Keevash · Discrete & Computational... · 0 citations

Preprint Jul 2026

On the largest size of sum-free sets in symmetric regions

A subset $S$ of a group $G$ is said to be sum-free (resp. $\Delta$-free) if there are no solutions to $a+b=c$ (resp. $a+b+c=0$) with $a,b,c\in S$. For a convex region $R\subset\mathbb{R}^d$, let $\sigma(R)$ denote the maximal proportion of the volume of $R$ that a sum-free subset of $R$ can occupy. We prove that $\sigma([-1,1]^d)=1/2$. Our proof employs a careful application of the Brunn-Minkowski inequality. Moreover, for the $d$-dimensional Euclidean ball $\mathbb{B}^d(0,1)$, we show that $\sigma(\mathbb{B}^d(0,1))\leq 1/2+o_d(1)$. We present two arguments for this. The first combines some routine harmonic analysis on the sphere with known bounds on values of the ultraspherical polynomials. The second more elementary argument proceeds by establishing that the maximal $\Delta$-free subset of the unit sphere $\mathbb{S}^{d-1}$ occupies $1/2+O(d^{-1})$ of the sphere's surface measure. This answers a question raised by Bukh.

Anubhab Ghosal, Dmitry Tsarev · 0 citations