2015/09/15 by Pablo Candela, Candela, Pablo, Balázs Szegedy +1 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1509.04485
32 pages. Referee's comments incorporated, yielding improvements in the exposition in sections 2.1, 5, and 6. To appear in Journal d'Analyse Mathématique
arxiv created 2016/09/11 · arxiv updated 2016/09/13
It is known that if p is a sufficiently large prime then for every function f:ℤp→ [0,1] there exists a continuous function on the circle f':\mathbbT→ [0,1] such that the averages of f and f' across any prescribed system of linear forms of complexity 1 differ by at most ε. 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 \mathbbT with the torus \mathbbT2 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 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 measurable functions f:\mathbbT2→ [0,1] with ∫_\mathbbT2f≥ α, of the following integral: ∫_\mathbbT5 f\binomx1y1 f\binomx1+x2y1+y2 f\binomx1+2x2y1+2y2+y3 f\binomx1+3 x2y1+3y2+3y3 dμ_\mathbbT5(x1,x2,y1,y2,y3).