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arXiv 1708.02613math.NTmath.DS
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乘性函数水平集的一个结构定理及其应用

A structure theorem for level sets of multiplicative functions and applications

Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Florian K. Richter

备注：
32 pages. Formerly part of arXiv:1705.07322
发表信息：
International Mathematical Research Notices, rny040 (2018)
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摘要：

给定一个任意乘法函数$f$的水平集$E$，在Frantzikinakis和Host [13，14]的基础上，我们建立了一个结构定理，给出了$\mathbb{1}_E$的概周期部分和伪随机部分的分解.利用这个结构定理和作者在[3]中发展的技巧，我们得到了下列关于多项式多重递归的结果。设$E=\{n_1<n_2<\ldots\}$是任意正密度乘性函数的水平集. - $E$是可除的，即集合$E\cap u\mathbb{N}$的上密度对于所有$u\in\mathbb{N}$都是正的;- $E$是多项式多重递归的平均集合，即对于所有测度保持系统$(X,\mathcal{B},μ,T)$、具有$μ(A)>0$的所有$A\in\mathcal{B}$、所有$\ell\geq 1$和所有多项式$p_i\in\mathbb{Z}[x]$、$i=1,\ldots,\ell$，我们还证明了，如果一个乘性函数的水平集$$ \lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N μ\big(A\cap T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$有正的上密度，则任何自移位$E-r$，$r\in E$，是一个平均多项式多重递归集。这反过来又导致了多项式Szemerédi定理的以下细化（参见。[4]）。设$E$是一个任意乘法函数的水平集，$E$有正的上密度，令$r\in E$。则对于任何具有正上密度的集合$D\subset \mathbb{N}$和任何多项式$p_i\in\mathbb{Q}[t]$，$i=1,\ldots,\ell$，只要对所有的$i\in\{1,\ldots,\ell\}$满足$p_i(\mathbb{Z})\subset\mathbb{Z}$和$p_i(0)=0$，就存在$β>0$使得集合$$ \left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>β\,\right\} $$具有正下密度.

英文摘要：

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost periodic and a pseudo-random parts. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. Let $E=\{n_1<n_2<\ldots\}$ be a level set of an arbitrary multiplicative function with positive density. Then the following are equivalent: - $E$ is divisible, i.e. the upper density of the set $E\cap u\mathbb{N}$ is positive for all $u\in\mathbb{N}$; - $E$ is an averaging set of polynomial multiple recurrence, i.e. for all measure preserving systems $(X,\mathcal{B},μ,T)$, all $A\in\mathcal{B}$ with $μ(A)>0$, all $\ell\geq 1$ and all polynomials $p_i\in\mathbb{Z}[x]$, $i=1,\ldots,\ell$, with $p_i(0)=0$ we have $$ \lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N μ\big(A\cap T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$ We also show that if a level set $E$ of a multiplicative function has positive upper density, then any self-shift $E-r$, $r\in E$, is a set of averaging polynomial multiple recurrence. This in turn leads to the following refinement of the polynomial Szemerédi theorem (cf. [4]). Let $E$ be a level set of an arbitrary multiplicative function, suppose $E$ has positive upper density and let $r\in E$. Then for any set $D\subset \mathbb{N}$ with positive upper density and any polynomials $p_i\in\mathbb{Q}[t]$, $i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $β>0$ such that the set $$ \left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>β\,\right\} $$ has positive lower density.

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