论文标题
模块化环中的furstenberg-sárközy定理和渐近的总牙齿现象
The Furstenberg-Sárközy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings
论文作者
论文摘要
furstenberg-sárközy定理断言，差异设置了一个子集$ e \ subset \ subset \ mathbb {n} $的$ e-e $与正密度的正相交，可与任何polynomial $ p \ in \ mathbb {z} [z} [n] $的图像集相交。 Furstenberg的方法依赖于对应原理和Poincaré复发定理的多项式版本，该定理源自Ergodic Wheoretic的结果，该结果对于任何措施提供系统$（x，x，\ nathcal {b}，μ，μ，t）$和set $ a \ in \ mathcal with \ n $ a $ a $ c $ c $ c $ c $ c $ cy（ \ lim_ {n \ to \ infty} \ frac {1} {n} \ sum_ {n = 1}^nμ（a \ cap t^{ - p（n）} a）>0。$ limit of $ c（a）$将具有$μ（a）^2 $的最佳值为$ t $的最佳值。在新组合应用的可能性中，我们在模块化环的设置$ \ mathbb {z}/n \ mathbb {z} $的情况下定义了渐近总呈现的概念。我们表明，一系列模块化环$ \ mathbb {z}/n_m \ mathbb {z} $，$ m \ in \ mathbb {n}中，$仅当$ andrm {lpf}（lpf}（lpf}（n_m）$，$ n__mm $ n __mmmm $ n __mmmm $ n __mmmm $ n __mmmm $ n __mmmm $ n __mmm $ n __mmm $ n_mmm $ nd__mmm $ n_mmm $ n __mmm $ n _ __mmmm $ nd__mmm $ n __mmmm $ n ___mmmm $ n_mmmm $从这个事实中，我们得出了一些组合后果，例如以下内容。修复$Δ\ in（0,1] $和a（不一定是相互作用的）polyenmial $ q \ in \ mathbb {q} [n] $，这样，$ q（\ supbb {z}）\ subseteq \ subseteq \ subseteq \ mathbb {z} $ \ MathBb {Z}/N \ MathBb {Z} \} $ $ \ mathbb {z}/n \ mathbb {z} = a + b + s $。
The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},μ,T)$ and set $A \in \mathcal{B}$ with $μ(A) > 0$, one has $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N μ(A \cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $μ(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings $\mathbb{Z}/N_m\mathbb{Z}$, $m \in \mathbb{N},$ is asymptotically totally ergodic if and only if $\mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $δ\in (0,1]$ and a (not necessarily intersective) polynomial $Q \in \mathbb{Q}[n]$ such that $Q(\mathbb{Z}) \subseteq \mathbb{Z}$, and write $S = \{ Q(n) : n \in \mathbb{Z}/N\mathbb{Z}\}$. For any integer $N > 1$ with $\mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $\mathbb{Z}/N\mathbb{Z}$ such that $|A||B| \geq δN^2$, then $\mathbb{Z}/N\mathbb{Z} = A + B + S$.
