Journal d'Analyse Mathématique

, Volume 135, Issue 2, pp 437–471

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

• Pablo Candela
• Balázs Szegedy
Article

## Abstract

It is known that if p is a sufficiently large prime, then, for every function f: Zp → [0, 1], there exists a continuous function f′: T → [0, 1] on the circle such that the averages of f and f′ across any prescribed system of linear forms of complexity 1 differ by at most ∈. 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 T with the torus T2 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 combine this notion with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on Zp. For instance, let m4(α, Zp) denote the minimum, over all sets A ⊆ Zp of cardinality at least αp, of the density of 4-term arithmetic progressions inside A. We show that limp→∞ m4(α, Zp) is equal to the infimum, over all continuous functions f: T2 →[0, 1] with $${\smallint _{{T^2}}}f \geqslant a$$, of the integral
$$\int_{{T^5}} {f\left( {\begin{array}{*{20}{c}} {{x_1}} \\ {{y_1}} \end{array}} \right)} f\left( {\begin{array}{*{20}{c}} {{x_1} + {x_2}} \\ {{y_1} + {y_2}} \end{array}} \right)f\left( {\begin{array}{*{20}{c}} {{x_1} + 2{x_2}} \\ {{y_1} + 2{y_2} + {y_3}} \end{array}} \right).f\left( {\begin{array}{*{20}{c}} {{x_1} + 3{x_2}} \\ {{y_1} + 3{y_2} = 3{y_3}} \end{array}} \right)d{\mu _{{T^5}}}({x_1},{x_2},{y_1},{y_2},{y_3})$$

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