arXiv subjects
Publications and source records attributed to Pavel Zorin-Kranich.
Let $G$ be a finite $D$-quasirandom group and $A \subset G^{k}$ a $δ$-dense subset. Then the density of the set of side lengths $g$ of corners \[ \{(a_{1},\dots,a_{k}),(ga_{1},a_{2},\dots,a_{k}),\dots,(ga_{1},\dots,ga_{k})\} \subset A \] converges to $1$ as $D\to\infty$.
We provide a versatile formulation of Lacey's recent sparse pointwise domination technique with a local weak type estimate on a nontangential maximal function as the only hypothesis. We verify this hypothesis for sharp variational truncations of singular integrals in the case when unweighted $L^2$ estimates are available. This extends previously known sharp weighted estimates for smooth variational truncations to the case of sharp variational truncations. We also include a sparse domination result for iterated commutators of multilinear operators with BMO functions.
We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.
Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0 < a < 1/2$, and let $p(n) = n^{1+ε}$, $0 < ε< 1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ the modulated, random averages \[ \frac{1}{N} \sum_{n = 1}^N e(p(n)) T^{a_n(ω)} f\] converge to $0$ pointwise almost everywhere.
We show that a $k$-linear pointwise ergodic theorem on an ergodic measure-preserving system implies a uniform $k$-linear nilsequence Wiener-Wintner theorem on that system. The assumption is known to hold for arbitrary systems and $k=2$ (due to Bourgain) and for distal systems and arbitrary $k$ (due to Huang, Shao, and Ye).
We show that the truncated simplex Hilbert transform enjoys some cancellation in the sense that its norm grows sublinearly in the number of scales retained in the truncation. This extends the recent result by Tao on cancellation for the multilinear Hilbert transform. Our main tool is the Hilbert space regularity lemma due to Gowers, which enables a very short proof.
The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well studied objects of harmonic analysis. We investigate $L^p$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one-dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.
We prove weighted and vector-valued variational estimates for ergodic averages on $\mathbb{R}^d$. The weighted square function estimate relating ergodic averages to the dyadic martingale is obtained using an $\ell^r$ version of a reverse Hölder inequality for variation seminorms.
We prove $q$-variation estimates, $q>2$, on $\ell^{p}$ spaces for averages along primes (with $1<p<\infty$) and polynomials (with $\big| \frac1p - \frac12 \big| < \frac{1}{2(d+1)}$, where $d$ is the degree of the polynomial). This improves the pointwise ergodic theorems for these averages in the corresponding ranges of $L^{p}$ spaces.
We prove, in a unified way, $r$-variational estimates, $r>2$, on $\ell^{s}(\mathbb{Z})$ spaces, $s \in (1, \infty)$, for averages and truncated singular integrals along the set of prime numbers.
We construct, in locally compact, second countable, amenable groups, sets with large density that fail to have certain combinatorial properties. For the property of being a shift of a set of measurable recurrence we show that this is possible when the group does not have cocompact von Neumann kernel. For the stronger property of being piecewise-syndetic we show that this is always possible. For minimally almost periodic, locally compact, second countable, amenable groups, we prove that any dilation of a positive density set by an open neighborhood of the identity contains a set of measurable recurrence, and that the same result holds, up to a shift, when the von Neumann kernel is cocompact. This leads to a trichotomy for locally compact, second countable, amenable groups based on combinatorial properties of large sets. We also prove, using a two-sided Furstenberg correspondence principle, that any two-sided dilation of a positive density set contains a two-sided finite products set.
Let $T_1$, $T_2$ be two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically. We show that $μ(A\cap T_1^g A\cap T_1^g T_2^g A) > μ(A)^4-ε$ on a syndetic set for any measurable set $A$ and any $ε>0$. The proof uses the concept of a sated system introduced by Austin.
We apply Walsh's method for proving norm convergence of multiple ergodic averages to arbitrary amenable groups. We obtain convergence in the uniform Cesàro sense for their polynomial actions and for ``triangular'' averages associated to commuting homomorphic actions. The latter generalizes a result due to Bergelson, McCutcheon, and Zhang in the case of two actions.
We extend Bourgain's return times theorem to arbitrary locally compact second countable amenable groups. The proof is based on a version of the Bourgain--Furstenberg--Katznelson--Ornstein orthogonality criterion.
The principal results proved in this expository thesis are the IP polynomial Szemerédi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence theorem for nilpotent polynomial multiple ergodic averages and the return times theorem to locally compact second countable amenable groups.
We give a new proof of Rudolph's multiple term return times theorem based on Host-Kra structure theory. Our approach provides characteristic factors for all terms, works for arbitrary tempered Følner sequences and also yields a multiple term Wiener-Wintner-type return times theorem for nilsequences.
We present the abstract framework and some applications of interpolation theory. The main new result concerns interpolation between H^1 and L^p estimates for analytic families of operators acting on Schwartz functions.
We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.