多项式增长Hardy序列的多重常返性和收敛性
Multiple recurrence and convergence for Hardy sequences of polynomial growth
摘要：
本文研究了多个遍历平均数的极限行为，这些平均数涉及满足某些正则性条件且具有多项式增长的整数序列。我们表明，“典型的”选择的Hardy场函数$a(t)$多项式增长，平均值$\frac{1}{N}\sum_{n=1}^N f_1(T^{[a(n)]}x)\cdot...\cdot f_\ell(T^{\ell [a(n)]}x)$收敛于平均值，我们确定其限制。例如，如果是$a(t)=t^{3/2}, t\log{t},$或$t^2+(\log{t})^2$，则是这种情况。此外，如果${a_1(t),...,a_\ell(t)}$是多项式增长的微指数函数的“典型”族，则对于每个遍历系统，平均值$\frac{1}{N}\sum_{n=1}^N f_1(T^{[a_1(n)]}x)\cdot...\cdot f_\ell(T^{[a_\ell(n)]}x)$在平均值中收敛于相应函数的积分的乘积。例如，如果函数$a_i(t)$由$t$的不同的正分数幂给出，则是这种情况。我们推导出组合数学中的几个结果。我们证明了，如果$a(t)$是一个非多项式的Hardy场函数的多项式增长，那么每一组正上密度的整数包含算术级数的形式${m,m+[a(n)],...,m+\ell[a(n)]}$。在适当的假设下，我们得到了关于${m, m+[a_1(n)],..., m+[a_\ell(n)]}.$形式的模式的一个相关结果
英文摘要：
We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for "typical" choices of Hardy field functions $a(t)$ with polynomial growth, the averages $\frac{1}{N}\sum_{n=1}^N f_1(T^{[a(n)]}x)\cdot...\cdot f_\ell(T^{\ell [a(n)]}x)$ converge in the mean and we determine their limit. For example, this is the case if $a(t)=t^{3/2}, t\log{t},$ or $t^2+(\log{t})^2$. Furthermore, if ${a_1(t),...,a_\ell(t)}$ is a "typical" family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages $\frac{1}{N}\sum_{n=1}^N f_1(T^{[a_1(n)]}x)\cdot...\cdot f_\ell(T^{[a_\ell(n)]}x)$ converge in the mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions $a_i(t)$ are given by different positive fractional powers of $t$. We deduce several results in combinatorics. We show that if $a(t)$ is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form ${m,m+[a(n)],...,m+\ell[a(n)]}$. Under suitable assumptions we get a related result concerning patterns of the form ${m, m+[a_1(n)],..., m+[a_\ell(n)]}.$
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