A structure theorem for multiple correlation sequences on nilsystems
Abstract
Recently, the author refuted several conjectures about the vanishing of multiple correlation sequences. The goal of this paper is to provide a counterpart for the conjectures of Frantzikinakis--Kuca and of Leibman. The significance of this manuscript is twofold: firstly, we provide a structure theorem for multiple correlation sequences on nilsystems along arithmetic progressions (in the sense of Frantzikinakis--Kuca), and a corresponding structure theorem for polynomial orbits of subnilmanifolds in connected nilmanifolds (in the sense of Leibman). Secondly, we identify null generalized nilsequences, as a notion of smallness, which provides a structured counterpart for null sequences. In particular, in the special case of ergodic nilsystems, we answer positively a question of Frantzikinakis on the structure of multiple correlation sequences.
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BibTeX
@misc{shalom2026structure,
title = {{A structure theorem for multiple correlation sequences on nilsystems}},
author = {Or Shalom},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33596},
eprint = {2610.33596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33596},
}
RIS
TY - UNPB
TI - A structure theorem for multiple correlation sequences on nilsystems
AU - Or Shalom
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33596
UR - https://arxiv.org/abs/2610.33596
AB - Recently, the author refuted several conjectures about the vanishing of multiple correlation sequences. The goal of this paper is to provide a counterpart for the conjectures of Frantzikinakis--Kuca and of Leibman. The significance of this manuscript is twofold: firstly, we provide a structure theorem for multiple correlation sequences on nilsystems along arithmetic progressions (in the sense of Frantzikinakis--Kuca), and a corresponding structure theorem for polynomial orbits of subnilmanifolds in connected nilmanifolds (in the sense of Leibman). Secondly, we identify null generalized nilsequences, as a notion of smallness, which provides a structured counterpart for null sequences. In particular, in the special case of ergodic nilsystems, we answer positively a question of Frantzikinakis on the structure of multiple correlation sequences.
ER -