arXiv 1509.044852015-09math.CO

简单阿贝尔群上复杂度为2的系统的连续模型

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

Pablo Candela, Balázs Szegedy

备注：: 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

摘要：

众所周知，如果$p$是一个足够大的素数，那么对于每个函数$f:\mathbb{Z}_p\to [0,1]$，在圆$f':\mathbb{T}\to [0,1]$上存在一个连续函数，使得在任何给定的复杂度为1的线性形式系统上，$f$和$f'$的平均值相差至多$ε$。这一结果如下工作的Sisask，建立在傅立叶分析参数的Croot回答了一个问题的绿色。我们将这一结果推广到系统的复杂性最多为2，更换$\mathbb{T}$与环面$\mathbb{T}^2$配备了一个特定的过滤。为此，我们使用一个概念的建模过滤nilmanifold，我们定义的equidistributed地图，我们结合这与工具的二次傅立叶分析。我们的研究结果产生的表达式上的环面的限制，涉及系统的复杂性2 $\mathbb{Z}_p$的组合量。例如，令$m_4(α,\mathbb{Z}_p)$表示在至少$αp$的基数的所有集合$A\subset \mathbb{Z}_p$上$A$内的4项算术级数的密度的最小值。我们证明$\lim_{p\to \infty} m_4(α,\mathbb{Z}_p)$等于下确界，在所有可测函数$f:\mathbb{T}^2\to [0,1]$与$\int_{\mathbb{T}^2}f\geq α$上，以下积分：$$ \int_{\mathbb{T}^5} f\binom{x_1}{y_1}\; f\binom{x_1+x_2}{y_1+y_2}\; f\binom{x_1+2x_2}{y_1+2y_2+y_3}\; f\binom{x_1+3 x_2}{y_1+3y_2+3y_3} \,dμ_{\mathbb{T}^5}(x_1,x_2,y_1,y_2,y_3). $$

英文摘要：

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 $ε$. 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 $\mathbb{T}$ 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 $m_4(α,\mathbb{Z}_p)$ denote the minimum, over all sets $A\subset \mathbb{Z}_p$ of cardinality at least $αp$, of the density of 4-term arithmetic progressions inside $A$. We show that $\lim_{p\to \infty} m_4(α,\mathbb{Z}_p)$ is equal to the infimum, over all measurable functions $f:\mathbb{T}^2\to [0,1]$ with $\int_{\mathbb{T}^2}f\geq α$, of the following integral: $$ \int_{\mathbb{T}^5} f\binom{x_1}{y_1}\; f\binom{x_1+x_2}{y_1+y_2}\; f\binom{x_1+2x_2}{y_1+2y_2+y_3}\; f\binom{x_1+3 x_2}{y_1+3y_2+3y_3} \,dμ_{\mathbb{T}^5}(x_1,x_2,y_1,y_2,y_3). $$

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