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Abstract:Let $\Gamma$ be a countable abelian group with Pontryagin dual $\Sigma:=\widehat{\Gamma}$. A non-commutative $2$-fold multiple correlation sequence is a double-indexed sequence $$b(\gamma,\gamma') = \int_X T_\gamma S_{\gamma'} f\cdot T_\gamma g\cdot h\,d\mu ,$$ where $f,h\in L^2(\mu)$ and $g\in L^\infty(\mu)$ are functions on a probability space $\XX=(X,\mathcal{B},\mu)$ and $T,S$ are two, not necessarily commuting, measure preserving $\Gamma$-actions. We prove that $b$ is of this form if and only if there are two finite Borel measures $\sigma_1,\sigma_2$ on $\Sigma$ and a bounded operator $G:L^2(\Sigma,\sigma_1)\rightarrow L^2(\Sigma,\sigma_2)$ with $$b(\gamma,\gamma')=\int_\Sigma G(\xi_\gamma)\cdot\xi_{\gamma'}\,d\sigma_2 ,$$ where $\xi_\gamma(\chi)=\chi(\gamma)$ is the evaluation map.
We then study the extent to which the triple $(G,\sigma_1,\sigma_2)$ is positive in the case $f=g=h=1_A$. Positivity fails in the sense available for the spectral measure of a single correlation sequence, but survives on positive functions with non-negative Fourier coefficients, where we also obtain a quantitative form by generalizing an inequality of Chu. As an application we prove a multiple recurrence theorem for products of linear forms, and deduce a simultaneous partition regularity result for a family of quadratic equations, extending, under a non-degeneracy hypothesis, a theorem of Frantzikinakis and Host.

Submission history

From: Or Shalom [view email]
[v1] Fri, 24 Feb 2023 19:14:58 UTC (45 KB)
[v2] Sun, 9 Aug 2026 16:56:23 UTC (61 KB)