On Bohman’s conjecture related to a sum packing problem of Erdos

Abstract

Motivated by a sum packing problem of Erdős, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let $H$ be a hyperplane in $\mathbb R^n$ such that $H\cap \{0,\pm 1\}^n=\{0^n\}$. The problem is to determine \[ f(n)\triangleq \max _H|H\cap \{0,\pm 1,\pm 2\}^n|.\] Bohman (1996) conjectured that \[ f(n)=\frac 12 (1+\sqrt 2)^n+\frac 12 (1-\sqrt 2)^n.\] We show that for some constants $c_1,c_2$ we have $c_1(2,538)^n<f(n)< c_2(2,723)^n$—disproving the conjecture. We also consider a more general question of the estimation of $|H\cap \{0,\pm 1,\dots ,\pm m\}|$, when $H\cap \{0,\pm 1,\dots ,\pm k\}=\{0^n\}$, $m>k>1$.

Locations

• Proceedings of the American Mathematical Society - View - PDF