Trace spectra of simplices in large sets

Abstract

Given an ordered tuple $\mathbf v=(v_0,\ldots,v_d)$ 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 $q\geq 1$, 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.