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Alexander Leibman

Publications and source records attributed to Alexander Leibman.

5 recordsLinked to original sources
arXiv2019

Nilsequences, null-sequences, and multiple correlation sequences

A (d-parameter) basic nilsequence is a sequence of the form ψ(n)=f(a^{n}x), n \in Z^{d}, where x is a point of a compact nilmanifold X, a is a translation on X, and f is a continuous function on X; a nilsequence is a uniform limit of basic nilsequences. If X is a compact nilmanifold, Y is a subnilmanifold of X, g(n) is a (d-parameter) polynomial sequence of translations of X, and f is a continuous function on X, we show that the sequence \int_{g(n)Y}f is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system (W,μ,T), integer polynomials p_{1},...,p_{k} on Z^{d}, and measurable sets A_{1},...,A_{k} in W, the sequence μ(T^{p_{1}(n)}A_{1}\cap...\cap T^{p_{k}(n)}A_{k}), n\in Z^{d}, is the sum of a nilsequence and a null-sequence.

math.DS↗
arXiv2014

Joint ergodicity along generalized linear functions

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{ϕ_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure preserving transformations of $X$ and $ϕ_{i}$ are integer valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending of a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{ϕ_{i}(n)}$ along primes.

math.DS↗
arXiv2007

Intersective polynomials and polynomial Szemeredi theorem

Let $P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}]$ be a family of polynomials such that $p_{i}(\Z^{m})\sle\Z$, $i=1,\ld,r$. We say that the family $P$ has {\it PSZ property} if for any set $E\sle\Z$ with $d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0$ there exist infinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression of the form \hbox{$\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}$}. We prove that a polynomial family $P=\{p_{1},\ld,p_{r}\}$ has PSZ property if and only if the polynomials $p_{1},\ld,p_{r}$ are {\it jointly intersective}, meaning that for any $k\in\N$ there exists $n\in\Z^{m}$ such that the integers $p_{1}(n),\ld,p_{r}(n)$ are all divisible by $k$. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If $p_{1},\ld,p_{r}\in\Q[n]$ are jointly intersective integral polynomials, then for any finite partition of $\Z$, $\Z=\bigcup_{i=1}^{k}E_{i}$, there exist $i\in\{1,\ld,k\}$ and $a,n\in E_{i}$ such that $\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}$.

math.DS↗
arXiv1999

Set-polynomials and polynomial extension of the Hales-Jewett Theorem

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established: Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a \subset V and a nonempty set γ\subseteq {1,...,N} such that a \cap (γ^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (γ^{d} x {1}), a \cup (γ^{d} x {2}), ..., a \cup (γ^{d} x {q}) are all of the same color. This ``polynomial'' Hales-Jewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.

math.CO↗