Structured extensions and multi-correlation sequences
James Leng
TL;DR
This work addresses the decomposition problem for k-fold multi-correlation sequences by proving that every such sequence can be written as the sum of a degree k generalized nilsequence and a null-sequence, solving a multidimensional version of a problem posed by Frantzikinakis. The authors develop a finitary inverse theorem and transfer its consequences to ergodic theory via the Furstenberg correspondence principle, yielding a structure theory for multidimensional Host-Kra factors and enabling a robust nil+null decomposition. Their approach blends combinatorial regularity (box norms and Taylor expansions), multi-scale nilsequence constructions, and a hierarchy of filtrations to bridge finitary and infinitary settings, with quasi-polynomial bounds. The results provide new tools for analyzing multidimensional ergodic averages and invite further work toward polynomial multi-correlation sequences and deeper extensions of Austin’s framework.
Abstract
We show that every multi-correlation sequence is the sum of a generalized nilsequence and a null-sequence. This proves a conjecture of N. Frantzikinakis. A key ingredient is the reduction of ergodic multidimensional inverse theorems to analogous finitary inverse theorems, offering a new approach to the structure theory of multidimensional Host-Kra factors. This reduction is proven by combining the methods of Tao (2015) with the Furstenberg correspondence principle. We also prove the analogous multidimensional finitary inverse theorem with quasi-polynomial bounds.