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.
A subset of the $d$-dimensional lattice cube $[n]^d$ is sum-free if it contains no solution to the equation $x+y=z$. We study the total number of such subsets. For $d=1$, Cameron and Erd\H{o}s conjectured that the number of sum-free subsets of $[n]$ is $O(2^{n/2})$, and this was proved independently by Green and Sapozh...
For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$...
Let $B^d$ denote the $d$-dimensional Euclidean ball of unit radius. What is the largest constant $f(B^d) \in [0,1]$ with the property that every finite collection $\mathcal{C}$ of unit balls in $\mathbb{R}^d$ admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $f(B^d)$ of the volume of $\mathca...
Let $p_c(\mathcal F)$, $q(\mathcal F)$, and $q_f(\mathcal F)$ denote the threshold, expectation threshold, and fractional expectation threshold of a family $\mathcal F$ of nonempty subsets of a finite set, respectively. We prove that there is an absolute constant $C>0$ such that, if $\mathcal F$ has VC dimension at mos...
We study the $(n-2)$-plane Kakeya problem in $\mathbb{R}^n$ ($n\geq4$) subject to convex Wolff axioms. In other words, we study finite families of $\delta$-neighborhoods of $(n-2)$-plane segments ($\delta$-flakes) in $\mathbb{R}^n$, with the property that not too many of them can be contained inside a common convex set...
Chun-Kit Lai, Yi-Xuan Pang, Rahul Sethi et al.· 0 citations
Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $\varphi\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $\varphi(z) = \varphi(x) \varphi(y)$. In 1990, Forcade and Pollington conjec...
John M. Campbell· 0 citations
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