Maximal polynomial modulations of singular integrals

Let $K$ be a standard H\"older continuous Calder\'on--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p(\mathbb{R}^{\mathbf{d}})$ for $1 < p < \infty$. This extends Sj\"olin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.Comment: v5: small corrections, more reference

A uniform nilsequence Wiener-Wintner theorem for bilinear ergodic averages

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).Comment: v2: 4 pages, characterization of good weights for L^2 convergence added, uniformity seminorm in the main result correcte

Cancellation for the simplex Hilbert transform

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.Comment: 8 page

Typical operators admit common cyclic vectors

Given a countable dense subset D of an infinite-dimensional separable Hilbert space H the set of operators for which every vector in D except zero is hypercyclic (weakly supercyclic) is residual for the strong (resp. weak) operator topology in the unit ball of L(H) multiplied by R>1 (resp. R>0)Comment: 7 pages, incorporating the referee's suggestion

A double return times theorem

We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$f_1(T^{a_1 n}x) f_2(T^{a_2 n}x)$$ is a good weight for the pointwise ergodic theorem.Comment: v2: 8 pages, improved typograph

Kakeya-Brascamp-Lieb inequalities

We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp-Lieb inequalities.Comment: v4: revised following referee reports, 18 page

Variation estimates for averages along primes and polynomials

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.Comment: v4: final, revised following referee's suggestion

Intrinsic square functions with arbitrary aperture

We consider intrinsic square functions defined using (log-)Dini continuous test functions on spaces of homogeneous type. We prove weighted estimates with optimal (at least in the Euclidean case) dependence on the aperture of the cone used to define the square function and linear dependence on the (log-)Dini modulus of continuity.Comment: v3: now on spaces of homogeneous type, 12 page

Corners over quasirandom groups

Let $G$ be a finite $D$-quasirandom group and $A \subset G^{k}$ a $\delta$-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 $D\to\infty$.Comment: 6 pages, with an expanded introductio

Cube spaces and the multiple term return times theorem

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 F{\o}lner sequences and also yields a multiple term Wiener-Wintner-type return times theorem for nilsequences.Comment: v2: 13 p., main result has been extended to tempered F{\o}lner sequence