论文标题
非惯性双线性多项式平均的尖角颈定理
Pointwise ergodic theorems for non-conventional bilinear polynomial averages
论文作者
论文摘要
我们在规范中建立了融合，几乎无处不在（在弗斯滕伯格）双线性多项式ergodic平均值\ [a_n（f，g）（x）：= \ frac {1} {n} {n} {n} \ sum_ { \ to \ to \ infty $，其中$ t \ colon x \ to x $是$σ$ -finite量度$（x，μ）$，$ p（\ mathrm {n}）\ in \ mathbb z [\ mathrm {n}] l^{p_1}（x），\ g \ in l^{p_2}（x）$，对于某些$ p_1，p_2> 1 $，带有$ \ frac {1} {1} {p_1} + \ frac {1} {1} {p_2} {p_2} {p_2} \ leq 1 $。我们还为这些平均值（在绝佳范围$ r> 2 $的lacunary量表上）建立了$ r $差异不平等。通过$ \ frac {1} {1} {p_1}+\ frac {1} {1} {p_2}> 1 $，以$ r $稍微增加$ r $，我们还可以通过$ \ frac {1} {p_1} {1} {1} {1} {p_1} {1} {1} {p_1} {1} {p_1} {1} {p_1} {p_1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {1} {p_2 $，以$ r $稍微增加$ r $来“打破双重性”。这给出了Frantzikinakis对Furstenberg的开放问题调查的问题11的肯定答案 - 我们的平均值（$ P（\ Mathrm {n}）= \ Mathrm {n}^2 $），这是Bergelson在他对Ergodic Ramsey The 1996的贡献中考虑的问题9的双线性变体。 Furstenberg-Bergelson-Leibman猜想。我们的方法将谐波分析中的技术与peluse和prendiville的最新定理相结合。在大尺度上，Adelic Integers $ \ MATHBB A _ {\ MATHBB Z} $的谐波分析也起着作用。
We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to \infty$, where $T \colon X \to X$ is a measure-preserving transformation of a $σ$-finite measure space $(X,μ)$, $P(\mathrm{n}) \in \mathbb Z[\mathrm{n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X), \ g \in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to "break duality" by handling some ranges of exponents $p_1,p_2$ with $\frac{1}{p_1}+\frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly. This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(\mathrm{n})=\mathrm{n}^2$), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. This also gives a contribution to the Furstenberg-Bergelson-Leibman conjecture. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $\mathbb A_{\mathbb Z}$ also plays a role.
