Trace spectra of simplices in large sets

Abstract

Given an ordered tuple $\mathbf{v}=(v_0,\dots ,v_d)$ of vectors in ${\mathbb{R}}^d$, let $A_{\mathbf{v}}=[v_1-v_0\cdots v_d-v_0]$ be its edge matrix. We prove that, in every finite colouring of ${\mathbb{R}}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \[ (c_2(A_{\mathbf v}),\ldots,c_d(A_{\mathbf v})). \] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\mathbf{v}})=\det (A_{\mathbf{v}})$. We also prove a discrete analogue: if $E\subseteq {\mathbb{Z}}^d$ has positive upper Banach density, then, for some $q\ge 1$, the set of coefficient tuples realized by ordered tuples in $E$ contains \[ q^2\mathbb{Z}\times q^3\mathbb{Z}\times\cdots\times q^d\mathbb{Z}. \] Finally, we show that the ordinary trace $c_1(A_{\mathbf{v}})$ cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices.

arXiv comment26 pages