论文标题
拓扑领域的多项式进程
Polynomial progressions in topological fields
论文作者
论文摘要
令$ p_1，\ ldots，p_m \在k [y] $中为多项式，在一般紧凑的拓扑字段$ k $中，没有恒定的术语和系数。我们给出了多项式进度的数量$ x，x + p_1（y），\ ldots，x + p_m（y）$，位于正密度的$ s \ subseteq k $中。证明依赖于具有独立利益的一般$ l^{\ infty} $ iNVERSE定理。这种逆定理意味着Sobolev改进了多线性多项式平均操作员的估计值，这反过来又意味着我们对多项式进行的定量估计值。这种普通的Sobolev不平等现象有可能在真实，复杂和$ p $ ADIC分析中的许多问题中应用。
Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots, x + P_m(y)$ lying in a set $S\subseteq K$ of positive density. The proof relies on a general $L^{\infty}$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and $p$-adic analysis.
