Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA decomposition of multicorrelation sequences for commuting transformations along primes
136
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $epsilon>0$, as the sum of a nilsequence $phi(n)$ and a sequence $omega(n)$ satisfying $lim_{Ntoinfty}frac{1}{N}sum_{n=1}^N |omega(n)|<epsilon$ and $lim_{Ntoinfty}frac{1}{|mathbb{P}cap [N]|}sum_{pin mathbb{P}cap [N]} |omega(p)|<epsilon$.
rate research
Read More
We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null sequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on $mathbb{Z}^{d}$-systems.
By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesins entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{N}^2$-actions.
We establish new recurrence and multiple recurrence results for a rather large family $mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $ellin mathbb{N}cup{0}$, $lim_{xtoinfty }f^{(ell)}(x)=pminfty$ and $lim_{xtoinfty }f^{(ell+1)}(x)=0$. Among other things, we show that for any $finmathcal{F}$, any invertible probability measure preserving system $(X,mathcal{B},mu,T)$, any $Ainmathcal{B}$ with $mu(A)>0$, and any $epsilon>0$, the sets of returns $$ R_{epsilon, A}= big{ninmathbb{N}:mu(Acap T^{-lfloor f(n)rfloor}A)>mu^2(A)-epsilonbig} $$ and $$ R^{(k)}_{A}= big{ ninmathbb{N}: mubig(Acap T^{lfloor f(n)rfloor}Acap T^{lfloor f(n+1)rfloor}Acapcdotscap T^{lfloor f(n+k)rfloor}Abig)>0big} $$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.
We use Maynards methods to show that there are bounded gaps between primes in the sequence ${lfloor nalpharfloor}$, where $alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $lfloor f(q)rfloor$ with $q$ a positive integer.
Consider averages along the prime integers $ mathbb P $ given by begin{equation*} mathcal{A}_N f (x) = N ^{-1} sum_{ p in mathbb P ;:; pleq N} (log p) f (x-p). end{equation*} These averages satisfy a uniform scale-free $ ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds begin{equation*} N ^{-1/p }lVert mathcal{A}_N frVert_{ell^{p}} leq C_p N ^{- 1/p} lVert frVert_{ell^p}. end{equation*} The maximal function $ mathcal{A}^{ast} f =sup_{N} lvert mathcal{A}_N f rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ mathcal{A}^{ast} $ is bounded on $ ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ mathcal{A}^{ast} $ were known.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
