Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences

TL;DR

This paper extends convergence results for the density of sets avoiding linear configurations in cyclic groups, using higher-order Fourier analysis and nilsequence regularity methods, applicable to general progressions and systems of linear forms.

Contribution

It generalizes convergence theorems for linear patterns in cyclic groups to all systems of finite complexity, employing periodic nilsequences and advanced regularity techniques.

Findings

Convergence of minimal progression counts for all k-term progressions.

Extension of convergence results to systems of linear equations.

Application of nilsequence regularity methods in finite cyclic groups.

Abstract

Given a positive integer $N$ and real number $\alpha\in [0, 1]$, let $m(\alpha,N)$ denote the minimum, over all sets $A\subset \mathbb{Z}/N\mathbb{Z}$ of size at least $\alpha N$, of the normalized count of 3-term arithmetic progressions contained in $A$. A theorem of Croot states that $m(\alpha,N)$ converges as $N\to\infty$ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for $k$-term progressions for general $k$ and further for all systems of integer linear forms of finite complexity. We also obtain a similar convergence result for the maximum densities of sets free of solutions to systems of linear equations. These results rely on a regularity method for functions on finite cyclic groups that we frame in terms of periodic nilsequences, using in particular some regularity results of Szegedy (relying on his joint work with Camarena) and equidistribution results of Green and Tao.