A continuous model for systems of complexity 2 on simple abelian groups

TL;DR

This paper extends the continuous modeling of functions from prime cyclic groups to the torus for systems of complexity 2, using quadratic Fourier analysis and filtered nilmanifolds, to analyze combinatorial structures like arithmetic progressions.

Contribution

It generalizes previous results from complexity 1 to complexity 2, introducing a new framework using filtered nilmanifolds and quadratic Fourier analysis for continuous models.

Findings

Established a limit expression for 4-term arithmetic progressions on lgebras of complexity 2

Developed a notion of modeling for filtered nilmanifolds based on equidistributed maps

Connected combinatorial quantities on lgebras to integrals over lgebras of complexity 2.

Abstract

It is known that if $p$ is a sufficiently large prime then for every function $f:\mathbb{Z}_p\to [0,1]$ there exists a continuous function on the circle $f':\mathbb{T}\to [0,1]$ such that the averages of $f$ and $f'$ across any prescribed system of linear forms of complexity 1 differ by at most $\epsilon$. This result follows from work of Sisask, building on Fourier-analytic arguments of Croot that answered a question of Green. We generalize this result to systems of complexity at most 2, replacing $\mathbb{T}$ with the torus $\mathbb{T}^2$ equipped with a specific filtration. To this end we use a notion of modelling for filtered nilmanifolds, that we define in terms of equidistributed maps, and we combine this with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on $\mathbb{Z}_p$. For instance, let $m_4(\alpha,\mathbb{Z}_p)$ denote the minimum, over all sets $A\subset \mathbb{Z}_p$ of cardinality at least $\alpha p$, of the density of 4-term arithmetic progressions inside $A$. We show that $\lim_{p\to \infty} m_4(\alpha,\mathbb{Z}_p)$ is equal to the infimum, over all measurable functions $f:\mathbb{T}^2\to [0,1]$ with $\int_{\mathbb{T}^2}f\geq \alpha$, of the following integral: $$ \int_{\mathbb{T}^5} f\binom{x_1}{y_1}\; f\binom{x_1+x_2}{y_1+y_2}\; f\binom{x_1+2x_2}{y_1+2y_2+y_3}\; f\binom{x_1+3 x_2}{y_1+3y_2+3y_3} \,d\mu_{\mathbb{T}^5}(x_1,x_2,y_1,y_2,y_3). $$