论文标题
沿算术进展的沿阵贡平均值降低度
Degree lowering for ergodic averages along arithmetic progressions
论文作者
论文摘要
我们检查了与算术渐进相关的多个千古平均值的限制行为，这些平均值是算法的差异是固定整数序列的元素。对于每个$ \ ell $，我们提供了必要和充分的条件，在上述表格的长度$ \ ell $的平均值中，其限制与$ \ ell $ term $ - term arithmetic进程的平均值相同。作为推论，我们得出了具有长度$ \ ell+1 $的算术进程的足够条件，并且在整数的密集子集中存在限制差异。这些结果是以下一般定理的结果：为了验证多个千古平均水平由$ d $ d $ gowers-host-kra seminorm控制，足以证明它由某些Gowers-host-kra seminorm控制，并且每当我们拥有$ d+d+d+d+1 $ 1 $ Control时，该学位$ d $。证明依赖于涉及双重功能的Gowers-host-KRA eminorms的基础定理，并结合了对双重函数符号平均值的新估计。我们使用这些估计值获得了先前用于涵盖积分乘积的平均值的降低参数的高阶变体。
We examine the limiting behavior of multiple ergodic averages associated with arithmetic progressions whose differences are elements of a fixed integer sequence. For each $\ell$, we give necessary and sufficient conditions under which averages of length $\ell$ of the aforementioned form have the same limit as averages of $\ell$-term arithmetic progressions. As a corollary, we derive a sufficient condition for the presence of arithmetic progressions with length $\ell+1$ and restricted differences in dense subsets of integers. These results are a consequence of the following general theorem: in order to verify that a multiple ergodic average is controlled by the degree $d$ Gowers-Host-Kra seminorm, it suffices to show that it is controlled by some Gowers-Host-Kra seminorm, and that the degree $d$ control follows whenever we have degree $d+1$ control. The proof relies on an elementary inverse theorem for the Gowers-Host-Kra seminorms involving dual functions, combined with novel estimates on averages of seminorms of dual functions. We use these estimates to obtain a higher order variant of the degree lowering argument previously used to cover averages that converge to the product of integrals.
