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Structure of multicorrelation sequences with integer part polynomial iterates along primes
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-10-16 , DOI: 10.1090/proc/15185
Andreas Koutsogiannis , Anh Le , Joel Moreira , Florian Richter

Abstract:Let $ T$ be a measure-preserving $ \mathbb{Z}^\ell $-action on the probability space $ (X,{\mathcal B},\mu ),$ let $ q_1,\dots ,q_m\colon \mathbb{R}\to \mathbb{R}^\ell $ be vector polynomials, and let $ f_0,\dots ,f_m \linebreak\in L^\infty (X)$. For any $ \epsilon > 0$ and multicorrelation sequences of the form $ \alpha (n) \linebreak =\int _Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu $ we show that there exists a nil-
sequence $ \psi $ for which $ \lim _{N - M \to \infty } \frac {1}{N-M} \sum _{n=M}^{N-1} \vert\alpha (n) - \psi (n)\vert \leq \epsilon $ and
$ \lim _{N \to \infty } \frac {1}{\pi (N)} \sum _{p \in \mathbb{P}\cap [1,N]} \vert\alpha (p) - \psi (p)\vert \leq \epsilon .$ This result simultaneously generalizes previous results of Frantzikinakis and the authors.


中文翻译:

整数部分多项式沿素数迭代的多重相关序列的结构

摘要:让我们在概率空间上$ T $采取措施,让它成为向量多项式,然后让。对于任何形式的和多重相关的序列,我们表明存在一个零序列,且该结果同时概括了Frantzikinakis和作者的先前结果。 $ \ mathbb {Z} ^ \ ell $ $(X,{\ mathcal B},\ mu),$ $ q_1,\ dots,q_m \冒号\ mathbb {R} \至\ mathbb {R} ^ \ ell $ $ f_0,\ dots,f_m \ linebreak \ in L ^ \ infty(X)$ $ \ epsilon> 0 $ $ \ alpha(n)\ linebreak = \ int _Xf_0 \ cdot T ^ {\ lfloor q_1(n)\ rfloor} f_1 \ cdots T ^ {\ lfloor q_m(n)\ rfloor} f_m \; d \ mu $
$ \ psi $ $ \ lim _ {N-M \ to \ infty} \ frac {1} {NM} \ sum _ {n = M} ^ {N-1} \ vert \ alpha(n)-\ psi(n)\ vert \ leq \ epsilon $
$ \ lim _ {N \ to \ infty} \ frac {1} {\ pi(N)} \ sum _ {p \ in \ mathbb {P} \ cap [1,N]} \ vert \ alpha(p) -\ psi(p)\ vert \ leq \ epsilon。$
更新日期:2020-11-12
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